How Many Parallel Sides Can A Triangle Have
A triangle is a geometric shape that always has three sides and three angles. Triangles have zero pairs of parallel lines.

Can a triangle have 2 parallel sides?

How to Find Parallel Lines on a Shape – To find parallel lines on a shape, extend each side of the shape with a ruler. If the lines cross, then the lines are not parallel. The lines are parallel if they do not cross. Alternatively, a ruler can be placed in line with one of the sides.

When moved to another side without rotation, the ruler will be in line with any other parallel sides. Horizontal and vertical lines are the easiest parallel lines to spot. These sides will be parallel to the edges of any paper or screens that they are displayed on. It is also helpful to be aware of common shapes that do and do not have parallel lines.

We saw above that a square has two pairs of parallel sides. A square is a special type of rectangle which has all sides the same length.

  1. Below is a rectangle.
  2. A rectangle has two pairs of parallel sides.
  3. One pair of parallel sides is horizontal and the other pair of parallel sides are vertical.

We label the first set of parallel sides with one arrow and the second pair of parallel sides with two arrows. Below is a isosceles trapezium. Isosceles trapeziums are symmetric.

  • We can look for parallel sides by extending the sides with a ruler.
  • Remember to look for horizontal and vertical sides.
  • We can see that there is a pair of parallel sides that both go horizontally from left to right.

We can extend the other two sides to check if they are parallel. We can see that these other two sides cross. Parallel lines do not cross and therefore, these two sides are not parallel.

  1. Here is a different trapezium.
  2. We first check any horizontal and vertical sides to see if they are parallel.
  3. We have a pair of vertical lines going from down to up.

Because they are both vertical, they are both parallel. We mark these parallel lines with an arrow. We can see that the other two sides cross when extended and so, they are not parallel sides. A trapezoid always has one pair of parallel sides. The shape below is called a parallelogram. It has the word parallel in its name. We can see that it has two horizontal sides, which are parallel. It also has another pair of parallel sides, which we mark with two arrows. A parallelogram has two pairs of parallel sides. Each side is always parallel to the side that is opposite to it.

  • A parallelogram is any four-sided shape that contains two pairs of parallel sides, where each side is parallel to the one opposite.
  • Therefore a rectangle, square and rhombus are all special kinds of parallelogram.
  • Many shapes do not have any parallel sides.
  • The shape in the example below is a regular pentagon, which has 5 sides that are all of the same length.

To check if there are any parallel lines on the sides of this shape, we can extend them with a ruler. If two lines cross then they are not parallel and they are parallel if they do not cross.

  1. We can see that no matter which two sides we take, extending the lines will always result in the lines crossing.
  2. No sides of this shape are parallel.
  3. Regular pentagons do not have any parallel sides.

Triangles are shapes that have no parallel sides. All triangles will have three sides and it is impossible for any of them to be parallel. If two lines out of three were parallel, it would be impossible to draw the shape. You can start by drawing any two parallel lines and then trying to complete the shape.

It is impossible to do by only using three sides in total. All types of triangle, such as equilateral triangle, isosceles triangle and scalene triangle, have no parallel lines. A kite is another shape that does not have parallel sides. Some shapes have many parallel sides. For example, a regular hexagon has three pairs of parallel sides.

5-Sided Square – Numberphile

Each side is parallel to the side that is opposite to it. A regular octagon is a shape that has 4 pairs of parallel sides. Each side is parallel to the side that is opposite to it. Now try our lesson on Sorting 2D Shapes where we learn how to sort shapes by their properties.

Can a triangle have 3 parallel sides?

None. Parallel lines are lines that will never cross each other, no matter how long you make them. A triangle is a polygon that has three sides and three angles. Due to the shape of a triangle, it has no parallel lines in it.

Can a triangle have two parallel sides True or false?

No, we cannot say that the two sides of a triangle are parallel. We can only say, for eg., a ║gram is parallel or a sq., rectangle and more. We cannot say a triangle is parallel because a side doesn’t have an opposite side. Saying the triangle is parallel is false.

What is the rule for parallel lines in a triangle?

If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. The converse is also true. In the image below, this tells us AD / DB = AE / EC.

Can a triangle have 2 perpendicular sides?

A right triangle has one right angle and two perpendicular lines.

Are parallel triangles equal?

Property: Equality of Areas of Triangles on Parallel Lines If two triangles lie on two parallel lines and they have bases of the same length, then they have the same area.

What shapes have 3 parallel sides?

The hexagon has three sets of parallel lines. This is the shape we were looking for.

Can a triangle have 1 2 3 as sides?

The answer is no. There are limits on what the lengths can be. For example, the lengths 1, 2, 3 cannot make a triangle because \begin 1 + 2 = 3\end, so they would all lie on the same line. The lengths 4, 5, 10 also cannot make a triangle because \begin 4 + 5 = 9\end.

Can 2 3 4 be sides of a triangle?

Yes, you can draw triangles with side lengths of 2 cm, 3 cm and 4 cm. According to the Triangles Inequality Theorm, the sum of two sides of a triangle should be greater than the third side. In this case sides are 2, 3 and 4. Hence our condition is satisfied and triangle can be formed.

What if two sides are parallel?

A quadrilateral whose two sides are parallel is known as a trapezium. True or false? Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses No worries! We‘ve got your back. Try BYJU‘S free classes today! : A quadrilateral whose two sides are parallel is known as a trapezium. True or false?

What if two sides are parallel and equal?

If one pair of opposite sides are equal and parallel, then the figure is,Q. Complete each of the following statements by means of one of those given in brackets against each : (i) If one pair of opposite sides is equal and parallel, then the figure is,

  1. Parallelogram, rectangle, trapezium) (ii) If in a quadrilateral only one pair of opposite sides is parallel, the quadrilateral is,
  2. Square, rectangle, trapezium) (iii) A line is drawn from the mid-point of one side of a triangle,, another side intersects the third side at its mid-point.

(perpendicular to, parallel to, to meet) (iv) If one angle of a parallelogram is a right angle, then it is necessarily a, (rectangle, square, rhombus) (v) The consecutive angle of a parallelogram is. (supplementary, complementary) (vi) If both pairs of opposite sides of a quadrilateral are equal, then it is necessarily a,

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What is the triangle sum theorem?

The triangle sum theorem (also known as the triangle angle sum theorem or angle sum theorem) states that the sum of the three interior angles of any triangle is always 180 degrees.

Is a rectangle a parallel?

The Improving Mathematics Education in Schools (TIMES) Project return to index Parallelograms and Rectangles Measurement and Geometry : Module 20 Years : 8-9 June 2011 PDF Version of module Assumed knowledge

Introductory plane geometry involving points and lines, parallel lines and transversals, angle sums of triangles and quadrilaterals, and general angle-chasing. The four standard congruence tests and their application in problems and proofs. Properties of isosceles and equilateral triangles and tests for them. Experience with a logical argument in geometry being written as a sequence of steps, each justified by a reason. Ruler-and-compasses constructions. Informal experience with special quadrilaterals.

Motivation There are only three important categories of special triangles − isosceles triangles, equilateral triangles and right-angled triangles. In contrast, there are many categories of special quadrilaterals. This module will deal with two of them − parallelograms and rectangles − leaving rhombuses, kites, squares, trapezia and cyclic quadrilaterals to the module, Rhombuses, Kites, and Trapezia,

  • Apart from cyclic quadrilaterals, these special quadrilaterals and their properties have been introduced informally over several years, but without congruence, a rigorous discussion of them was not possible.
  • Each congruence proof uses the diagonals to divide the quadrilateral into triangles, after which we can apply the methods of congruent triangles developed in the module, Congruence,

The present treatment has four purposes:

The parallelogram and rectangle are carefully defined. Their significant properties are proven, mostly using congruence. Tests for them are established that can be used to check that a given quadrilateral is a parallelogram or rectangle − again, congruence is mostly required. Some ruler-and-compasses constructions of them are developed as simple applications of the definitions and tests.

The material in this module is suitable for Year 8 as further applications of congruence and constructions. Because of its systematic development, it provides an excellent introduction to proof, converse statements, and sequences of theorems. Considerable guidance in such ideas is normally required in Year 8, which is consolidated by further discussion in later years.

The complementary ideas of a ‘property’ of a figure, and a ‘test’ for a figure, become particularly important in this module. Indeed, clarity about these ideas is one of the many reasons for teaching this material at school. Most of the tests that we meet are converses of properties that have already been proven.

For example, the fact that the base angles of an isosceles triangle are equal is a property of isosceles triangles. This property can be re-formulated as an ‘If, then ‘ statement:

If two sides of a triangle are equal, then the angles opposite those sides are equal.

Now the corresponding test for a triangle to be isosceles is clearly the converse statement:

If two angles of a triangle are equal, then the sides opposite those angles are equal.

Remember that a statement may be true, but its converse false. It is true that ‘If a number is a multiple of 4, then it is even’, but it is false that ‘If a number is even, then it is a multiple of 4′. Content Quadrilaterals In other modules, we defined a quadrilateral to be a closed plane figure bounded by four intervals, and a convex quadrilateral to be a quadrilateral in which each interior angle is less than 180°. We proved two important theorems about the angles of a quadrilateral:

The sum of the interior angles of a quadrilateral is 360°. The sum of the exterior angles of a convex quadrilateral is 360°.

To prove the first result, we constructed in each case a diagonal that lies completely inside the quadrilateral. This divided the quadrilateral into two triangles, each of whose angle sum is 180°. To prove the second result, we produced one side at each vertex of the convex quadrilateral. A parallelogram is a quadrilateral whose opposite sides are parallel. Thus the quadrilateral ABCD shown opposite is a parallelogram because AB || DC and DA || CB, The word ‘parallelogram’ comes from Greek words meaning ‘parallel lines’. Constructing a parallelogram using the definition To construct a parallelogram using the definition, we can use the copy-an-angle construction to form parallel lines. This is not the easiest way to construct a parallelogram. First property of a parallelogram − The opposite angles are equal The three properties of a parallelogram developed below concern first, the interior angles, secondly, the sides, and thirdly the diagonals. The first property is most easily proven using angle-chasing, but it can also be proven using congruence. Theorem The opposite angles of a parallelogram are equal. Proof

Let ABCD be a parallelogram, with A = α and B = β,
Prove that C = α and D = β.
α + β = 180° (co-interior angles, AD || BC ),
so C = α (co-interior angles, AB || DC )
and D = β (co-interior angles, AB || DC ).

Second property of a parallelogram − The opposite sides are equal As an example, this proof has been set out in full, with the congruence test fully developed. Most of the remaining proofs however, are presented as exercises, with an abbreviated version given as an answer. Theorem The opposite sides of a parallelogram are equal. Proof

ABCD is a parallelogram.
To prove that AB = CD and AD = BC,
J oin the diagonal AC,
In the triangles ABC and CDA :
BAC = DCA (alternate angles, AB || DC )
BCA = DAC (alternate angles, AD || BC )
AC = CA (common)
so ABC ≡ CDA (AAS)
Hence AB = CD and BC = AD (matching sides of congruent triangles).

Third property of a parallelogram − The diagonals bisect each other Theorem The diagonals of a parallelogram bisect each other. EXERCISE 1 a Prove that ABM ≡ CDM, b Hence prove that the diagonals bisect each other. As a consequence of this property, the intersection of the diagonals is the centre of two concentric circles, one through each pair of opposite vertices. Notice that, in general, a parallelogram does not have a circumcircle through all four vertices. First test for a parallelogram − The opposite angles are equal Besides the definition itself, there are four useful tests for a parallelogram. Second test for a parallelogram − Opposite sides are equal This test is the converse of the property that the opposite sides of a parallelogram are equal. Theorem If the opposite sides of a (convex) quadrilateral are equal, then the quadrilateral is a parallelogram. EXERCISE 3 Prove this result using congruence in the figure to the right, where the diagonal AC has been joined. This test gives a simple construction of a parallelogram given two adjacent sides − AB and AD in the figure to the right. Draw a circle with centre B and radius AD, and another circle with centre D and radius AB, The circles intersect at two points − let C be the point of intersection within the non-reflex angle BAD, Then ABCD is a parallelogram because its opposite sides are equal. It also gives a method of drawing the line parallel to a given line through a given point P, Choose any two points A and B on, and complete the parallelogram PABQ, Then PQ || Third test for a parallelogram − One pair of opposite sides are equal and parallel This test turns out to be very useful, because it uses only one pair of opposite sides. Theorem If one pair of opposite sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram. EXERCISE 4 Complete the proof using the figure on the right. This test for a parallelogram gives a quick and easy way to construct a parallelogram using a two-sided ruler. Draw a 6 cm interval on each side of the ruler. Joining up the endpoints gives a parallelogram. The test is particularly important in the later theory of vectors. Suppose that and are two directed intervals that are parallel and have the same length − that is, they represent the same vector. Then the figure ABQP to the right is a parallelogram. Even a simple vector property like the commutativity of the addition of vectors depends on this construction. The parallelogram ABQP shows, for example, that + = = + Fourth test for a parallelogram − The diagonals bisect each other This test is the converse of the property that the diagonals of a parallelogram bisect each other. Theorem If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram: EXERCISE 5 Complete the proof using the diagram below. This test gives a very simple construction of a parallelogram. Draw two intersecting lines, then draw two circles with different radii centred on their intersection. Join the points where alternate circles cut the lines. This is a parallelogram because the diagonals bisect each other. It also allows yet another method of completing an angle BAD to a parallelogram, as shown in the following exercise. EXERCISE 6 Given two intervals AB and AD meeting at a common vertex A, construct the midpoint M of BD, Complete this to a construction of the parallelogram ABCD, justifying your answer. Parallelograms Definition of a parallelogram A parallelogram is a quadrilateral whose opposite sides are parallel. Properties of a parallelogram

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The opposite angles of a parallelogram are equal. The opposite sides of a parallelogram are equal. The diagonals of a parallelogram bisect each other.

Tests for a parallelogram A quadrilateral is a parallelogram if:

its opposite angles are equal, or its opposite sides are equal, or one pair of opposite sides are equal and parallel, or its diagonals bisect each other.

Rectangles The word ‘rectangle’ means ‘right angle’, and this is reflected in its definition. Definition of a Rectangle A rectangle is a quadrilateral in which all angles are right angles. First Property of a rectangle − A rectangle is a parallelogram Each pair of co-interior angles are supplementary, because two right angles add to a straight angle, so the opposite sides of a rectangle are parallel. This means that a rectangle is a parallelogram, so:

Its opposite sides are equal and parallel. Its diagonals bisect each other.

Second property of a rectangle − The diagonals are equal The diagonals of a rectangle have another important property − they are equal in length. The proof has been set out in full as an example, because the overlapping congruent triangles can be confusing. Theorem The diagonals of a rectangle are equal. Proof Let ABCD be a rectangle. We prove that AC = BD, In the triangles ABC and DCB :

BC = CB (common)
AB = DC (opposite sides of a parallelogram)
ABC = DCA = 90° (given)

so ABC ≡ DCB (SAS) Hence AC = DB (matching sides of congruent triangles). This means that AM = BM = CM = DM, where M is the intersection of the diagonals. Thus we can draw a single circle with centre M through all four vertices. We can describe this situation by saying that, ‘The vertices of a rectangle are concyclic’. EXERCISE 7 Give an alternative proof of this result using Pythagoras’ theorem.

First test for a rectangle − A parallelogram with one right angle If a parallelogram is known to have one right angle, then repeated use of co-interior angles proves that all its angles are right angles. Theorem If one angle of a parallelogram is a right angle, then it is a rectangle. Because of this theorem, the definition of a rectangle is sometimes taken to be ‘a parallelogram with a right angle’.

Construction of a rectangle We can construct a rectangle with given side lengths by constructing a parallelogram with a right angle on one corner. First drop a perpendicular from a point P to a line, Mark B and then mark off BC and BA and complete the parallelogram as shown below. Second test for a rectangle − A quadrilateral with equal diagonals that bisect each other We have shown above that the diagonals of a rectangle are equal and bisect each other. Conversely, these two properties taken together constitute a test for a quadrilateral to be a rectangle. Theorem A quadrilateral whose diagonals are equal and bisect each other is a rectangle. EXERCISE 8 a Why is the quadrilateral a parallelogram? b Use congruence to prove that the figure is a rectangle. EXERCISE 9 Give an alternative proof of the theorem using angle-chasing. As a consequence of this result, the endpoints of any two diameters of a circle form a rectangle, because this quadrilateral has equal diagonals that bisect each other. Thus we can construct a rectangle very simply by drawing any two intersecting lines, then drawing any circle centred at the point of intersection. Rectangles Definition of a rectangle A rectangle is a quadrilateral in which all angles are right angles. Properties of a rectangle

A rectangle is a parallelogram, so its opposite sides are equal. The diagonals of a rectangle are equal and bisect each other.

Tests for a rectangle

A parallelogram with one right angle is a rectangle. A quadrilateral whose diagonals are equal and bisect each other is a rectangle.

Links forward The remaining special quadrilaterals to be treated by the congruence and angle-chasing methods of this module are rhombuses, kites, squares and trapezia. The sequence of theorems involved in treating all these special quadrilaterals at once becomes quite complicated, so their discussion will be left until the module Rhombuses, Kites, and Trapezia,

Each individual proof, however, is well within Year 8 ability, provided that students have the right experiences. In particular, it would be useful to prove in Year 8 that the diagonals of rhombuses and kites meet at right angles − this result is needed in area formulas, it is useful in applications of Pythagoras’ theorem, and it provides a more systematic explanation of several important constructions.

The next step in the development of geometry is a rigorous treatment of similarity. This will allow various results about ratios of lengths to be established, and also make possible the definition of the trigonometric ratios. Similarity is required for the geometry of circles, where another class of special quadrilaterals arises, namely the cyclic quadrilaterals, whose vertices lie on a circle.

Special quadrilaterals and their properties are needed to establish the standard formulas for areas and volumes of figures. Later, these results will be important in developing integration. Theorems about special quadrilaterals will be widely used in coordinate geometry. Rectangles are so ubiquitous that they go unnoticed in most applications.

One special role worth noting is they are the basis of the coordinates of points in the cartesian plane − to find the coordinates of a point in the plane, we complete the rectangle formed by the point and the two axes. Parallelograms arise when we add vectors by completing the parallelogram − this is the reason why they become so important when complex numbers are represented on the Argand diagram.

History and applications Rectangles have been useful for as long as there have been buildings, because vertical pillars and horizontal crossbeams are the most obvious way to construct a building of any size, giving a structure in the shape of a rectangular prism, all of whose faces are rectangles. The diagonals that we constantly use to study rectangles have an analogy in building − a rectangular frame with a diagonal has far more rigidity than a simple rectangular frame, and diagonal struts have always been used by builders to give their building more strength.

Parallelograms are not as common in the physical world (except as shadows of rectangular objects). Their major role historically has been in the representation of physical concepts by vectors. For example, when two forces are combined, a parallelogram can be drawn to help compute the size and direction of the combined force. In the triangles ABM and CDM :

1. BAM = DCM (alternate angles, AB || DC )
2. ABM = CDM (alternate angles, AB || DC )
3. AB = CD (opposite sides of parallelogram ABCD )
ABM = CDM (AAS)

b Hence AM = CM and DM = BM (matching sides of congruent triangles) EXERCISE 2

From the diagram, 2 α + 2 β = 360 o (angle sum of quadrilateral ABCD )
α + β = 180 o

table>

Hence AB || DC (co-interior angles are supplementary) and AD || BC (co-interior angles are supplementary).

EXERCISE 3

First show that ABC ≡ CDA using the SSS congruence test.
Hence ACB = CAD and CAB = ACD (matching angles of congruent triangles)
so AD || BC and AB || DC (alternate angles are equal.)

EXERCISE 4

First prove that ABD ≡ CDB using the SAS congruence test.
Hence ADB = CBD (matching angles of congruent triangles)
so AD || BC (alternate angles are equal.)

EXERCISE 5

First prove that ABM ≡ CDM using the SAS congruence test.
Hence AB = CD (matching sides of congruent triangles)
Also ABM = CDM (matching angles of congruent triangles)
so AB || DC (alternate angles are equal):

Hence ABCD is a parallelogram, because one pair of opposite sides are equal and parallel. EXERCISE 6 Join AM, With centre M, draw an arc with radius AM that meets AM produced at C, Then ABCD is a parallelogram because its diagonals bisect each other.

a We have already proven that a quadrilateral whose diagonals bisect each other is a parallelogram.

table>

b Because ABCD is a parallelogram, its opposite sides are equal. Hence ABC ≡ DCB (SSS) so ABC = DCB (matching angles of congruent triangles). But ABC + DCB = 180 o (co-interior angles, AB || DC ) so ABC = DCB = 90 o,

Hence ABCD is rectangle, because it is a parallelogram with one right angle. EXERCISE 9

ADM = α (base angles of isosceles ADM )
and ABM = β (base angles of isosceles ABM ),
so 2 α + 2 β = 180 o (angle sum of ABD )
α + β = 90 o,

Hence A is a right angle, and similarly, B, C and D are right angles. The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations. The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Education, Employment and Workplace Relations.

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© The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE-EM), the education division of the Australian Mathematical Sciences Institute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

https://creativecommons.org/licenses/by-nc-nd/3.0/

What if a triangle has 2 equal sides?

An isosceles triangle is a triangle with (at least) two equal sides. In the figure above, the two equal sides have length and the remaining side has length This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles. The name derives from the Greek iso (same) and skelos ( leg ). A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is called a scalene triangle,

(1)

The area is therefore given by The inradius of an isosceles triangle is given by

(5)

The mean of is given by so the geometric centroid is or 2/3 the way from its vertex (Gearhart and Schulz 1990). Considering the angle at the apex of the triangle and writing instead of, there is a surprisingly simple relationship between the area and vertex angle, As shown in the above diagram, simple trigonometry gives so the area is Erecting similar isosceles triangles on the edges of an initial triangle gives another triangle such that,, and concur. The triangles are therefore perspective triangles, No set of 6″> points in the plane can determine only isosceles triangles.

Can a triangle have 2 lines of symmetry?

Can you draw a triangle, which has exactly two lines of symmetry? Sketch a rough figure. Join Vedantu’s FREE Mastercalss Answer Verified Hint: In this question, we have to draw a triangle, which has exactly two lines of symmetry. For this, we will draw three types of triangles based on sides, which are the scalene triangle, isosceles triangle, and equilateral triangle.

After that, we will try to draw lines of symmetry of these triangles and check if any of them have exactly two lines of symmetry. Complete step-by-step solution Here, we have to draw a triangle, which has exactly two lines of symmetry.Let us first understand what a line of symmetry is.A line of symmetry is a line that cuts a shape exactly in half.

If we place a mirror along the line, the shape will remain unchanged.Based on this, we can draw three triangles based on their sides which are the scalene triangle, isosceles triangle, and equilateral triangle. Now, let us draw these triangles and their possible lines of symmetry.Scalene triangle: It is a triangle, which has all three sides different. As we can see, the scalene triangle has no line of symmetry. No line can cut this triangle into exactly two identical parts. So this is not our required triangle.Isosceles triangle: It is a triangle, which has two sides equal. (AB = AD in triangle ABC). As we can see, this triangle has only one line of symmetry. So, this is not our required triangle.Equilateral triangle: It is a triangle which has all sides equal. (AB = AC = BC in triangle ABC). As we can see, this triangle has exactly three lines of symmetry. So this is not our required triangle. Hence, there exists no triangle with exactly two lines of symmetry. Note: Students should take care that the line of symmetry creates two shapes which are mirror images of each other.

What if a triangle has two same sides?

An isosceles triangle is a triangle with two equal sides. An isosceles triangle, therefore, has both two equal sides and two equal angles.

Are parallel sides always equal?

Parallel sides of a quadrilateral, or any polygon, must be straight sides. They can be sides of equal length, but they do not have to be.

Are parallel always equal?

Are Parallel Lines Equal in Length? – No, parallel lines may not be equal in length but they should be the same distance apart.

Do parallel sides have equal angles?

Properties of Parallel Lines – As we have already learned, if two lines are parallel, they do not intersect, on a common plane. Now if a transversal intersects two parallel lines, at two distinct points, then there are four angles formed at each point. Hence, below are the properties of parallel lines with respect to transversals.

  • Corresponding angles are equal.
  • Vertical angles/ Vertically opposite angles are equal.
  • Alternate interior angles are equal.
  • Alternate exterior angles are equal.
  • Pair of interior angles on the same side of the transversal are supplementary.

Can a triangle have 2 lines of symmetry?

Can you draw a triangle, which has exactly two lines of symmetry? Sketch a rough figure. Join Vedantu’s FREE Mastercalss Answer Verified Hint: In this question, we have to draw a triangle, which has exactly two lines of symmetry. For this, we will draw three types of triangles based on sides, which are the scalene triangle, isosceles triangle, and equilateral triangle.

  1. After that, we will try to draw lines of symmetry of these triangles and check if any of them have exactly two lines of symmetry.
  2. Complete step-by-step solution Here, we have to draw a triangle, which has exactly two lines of symmetry.Let us first understand what a line of symmetry is.A line of symmetry is a line that cuts a shape exactly in half.

If we place a mirror along the line, the shape will remain unchanged.Based on this, we can draw three triangles based on their sides which are the scalene triangle, isosceles triangle, and equilateral triangle. Now, let us draw these triangles and their possible lines of symmetry.Scalene triangle: It is a triangle, which has all three sides different. As we can see, the scalene triangle has no line of symmetry. No line can cut this triangle into exactly two identical parts. So this is not our required triangle.Isosceles triangle: It is a triangle, which has two sides equal. (AB = AD in triangle ABC). As we can see, this triangle has only one line of symmetry. So, this is not our required triangle.Equilateral triangle: It is a triangle which has all sides equal. (AB = AC = BC in triangle ABC). As we can see, this triangle has exactly three lines of symmetry. So this is not our required triangle. Hence, there exists no triangle with exactly two lines of symmetry. Note: Students should take care that the line of symmetry creates two shapes which are mirror images of each other.

What if a triangle has two same sides?

An isosceles triangle is a triangle with two equal sides. An isosceles triangle, therefore, has both two equal sides and two equal angles.

What if a triangle has 2 equal sides?

An isosceles triangle is a triangle with (at least) two equal sides. In the figure above, the two equal sides have length and the remaining side has length This property is equivalent to two angles of the triangle being equal. An isosceles triangle therefore has both two equal sides and two equal angles. The name derives from the Greek iso (same) and skelos ( leg ). A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is called a scalene triangle,

(1)

The area is therefore given by The inradius of an isosceles triangle is given by

(5)

The mean of is given by so the geometric centroid is or 2/3 the way from its vertex (Gearhart and Schulz 1990). Considering the angle at the apex of the triangle and writing instead of, there is a surprisingly simple relationship between the area and vertex angle, As shown in the above diagram, simple trigonometry gives so the area is Erecting similar isosceles triangles on the edges of an initial triangle gives another triangle such that,, and concur. The triangles are therefore perspective triangles, No set of 6″> points in the plane can determine only isosceles triangles.

What does it mean if two sides are parallel?

A geometric shape has parallel sides if those sides do not meet or cross and the distance between them does not change. Parallel sides of a shape are opposite, or across from each other, and they would not intersect if they were extended infinitely beyond the confines of the shape.