## Envision Math Grade 8 Volume 1 Chapter 6 Congruence And Similarity

**Page 359 Exercise 1 Answer**

The figure of triangles which represent each flag is shown below.

The figure of triangles which represent each flag is:

We have to defined as, How are the side lengths of the triangles related?

The figure of triangles which represent each flag is shown below.

The figure of triangles which represent each flag is:

Given – The angles of both triangles are similar.

To do – How the measurements are related?

From the picture it is clearly visible that, the larger triangle have the same angle measure as the smaller triangle.

We can conclude from properties of triangles that angles are congruent.

The angles are congruent.

**Page 359 Focus On Math Practices Answer**

Given – Justin makes a third flag that has sides that are shorter than the sides of the small flag. Two of the angles for each flag measure the same.

To Find – prove that if the third angle of each flag have same measure.

By making the third flag which have sides smaller than the sides of small flag, the transformation method dilation is used.

By making the third flag which have sides smaller than the sides of small flag, the transformation method dilation is used.

From the properties of this transformation, it is known that corresponding angle measurement remains the same.

We can conclude that the third angle for each flag have same measurements.

**Page 360 Try It Answer**

Given – ∠Y = 92^{∘},∠M = 92^{∘},∠Z = 42^{∘},∠L = 53^{∘}

To do – Find m ∠X, m ∠N

**Step 1 of 4**

For determining whether the given triangles are similar, we will use the Angle-Angle (AA) Criterion

The angles are similar if two angles of a triangle are congruent to the corresponding angles of another triangle.

Therefore, we have to find the missing angle measure first:

Sum of the measures of the interior angles of a triangle is 180^{∘}.

**Step 2 of 4**

**Step 3 of 4**

m ∠N:

**Step 4 of 4**

So, we get that two corresponding angles are not congruent:

∠X ≠ ∠L, ∠Y ≅ ∠M, ∠Z ≠ ∠N

Using the Angle-Angle (AA) Criterion, we can conclude that triangle are not similar.

The answers are

46^{∘}

35^{∘}

The triangles are not similar.

**Page 361 Try It Answer**

Given – Knowledge of transformations and parallel lines.

To do – Explain that why angle-angle criterion is true for all triangles.

Step 1 of 1

If QR ∥ YZ

Step 2 of 2

Then, ∠1 ≅ ∠5 ∠2 ≅ ∠6

Because, these pair of angles are alternate interior angles.

Moreover, since ΔXYZ and ΔXRQ are isosceles triangles, the true is:

∠1 ≅ ∠2 ∠5 ≅ ∠6

Therefore:

m∠1 = m∠2 = m∠5 = m∠6

Besides, m∠3 = m∠4, because that is the same angle.

So, using the Angle-Angle Criterion, we can conclude that the given triangles are similar.

ΔXYZ ∼ ΔXRQ

ΔXYZ ∼ ΔXRQ

**Page 362 Exercise 1 Answer**

To do – Using angle measures to determine whether two triangles are similar.

We will use Angle-Angle (AA) Criterion of triangles

Step 1 of 1

There are some rule called Angle-Angle (AA) Criterion.

This criterion states that if two angles in one triangle are congruent to two angles in another triangle, the two triangles are similar triangles.

Angle-Angle criterion measures will be used to determine whether two triangles are similar.

**Page 362 Exercise 3 Answer**

Given: Three pairs of triangles

Two right triangles

Two isosceles right triangles

Two equilateral triangles

To find: Which triangle pairs below are always similar

For this we will apply AA similarity criterion to check which pair is always similar

If we have two right angled triangles then definitely 1 pair of angles which is 900 is always same, but the other two angles can vary. Hence not all right triangles are similar, or two right

In isosceles right triangles, the three angles are always fixed. One angle will be a right angle with 90°measure, and the other two angles will always be 45° and 45°(since it is an isosceles triangle). Hence, any two isosceles right triangles will have the same angles, satisfying AA criterion; and thus will always be similar.

Measure of the angles of any equilateral triangle is fixed i.e. each angle is of 600. Hence any 2 equilateral triangles will follow the AA similarity criterion. Therefore two equilateral triangles are always similar.

Two equilateral triangles and two isosceles right triangles are always similar, whereas two right angled triangles are not always similar

**Page 362 Exercise 4 Answer**

Given: Two triangles in which first triangle has angles 44° and 46° and the second triangle has two angles 90° and 46°

To find: Whether the two triangles are similar or not

We will find all angles of the given triangles and check whether they follow the AA similarity criterion or not

In the given triangle the two angles are 44°

and 46°. Let the third angle be x.

Now, 44° + 46° + x = 180°(Angle sum property of triangle)

90° + x = 180°

x = 180° − 90° = 90°

Hence the third angle of the triangle is 90°

Clearly the two triangles have two equal angles i.e. 46° and 90°

hence AA similarity criterion can be applied and therefore the two triangles are similar

The given two triangles are similar to each other

**Page 362 Exercise 5 Answer**

Given: Two triangles QLM and QRS with R lying on QL and S lying on QM.

∠QLM = ∠QRS = 90°

To find: Whether ΔQLM ∼ ΔQRS

We will try and apply AA similarity criterion to check the two triangles are similar or not.

In ΔQLM and ΔQRS

∠QLM = ∠QRS = 90° (Given)

∠Q = ∠Q (Common Angle)

Hence by AA similarity rule

ΔQLM ∼ ΔQRS

Hence ΔQLM ∼ ΔQRS

**Page 362 Exercise 6 Answer**

Given: Two triangles on same base two angles of one of the triangle are 560 and 760 and of the second triangle are 48° and 76°

To find:

Whether the two triangles are similar or not

Value of x

We will apple the property of The measure of the exterior angle of a triangle is equal to the sum of the remote interior angles.

Then we will check whether AA similarity criterion is applicable or not.

Let the third angle of the triangle with two angles 56° and 76° be y

Then, 56° + 76° + y = 180° (Sum of the three angles of a triangle is 180°)

y = 180 − 56 − 76 = 48°

Hence the three angles are 56°,76° and 48°

As both the triangles have two corresponding angles i.e. 48° and 76° equal hence by AA similarity rule we can say that the two triangles are similar to each other.

56° + 76° = 4x + 48° (The measure of the exterior angle of a triangle is equal to the sum of the remote interior angles.)

The two triangles are similar to each other the value of

x = 21°

**Page 363 Exercise 7 Answer**

Given: Two triangles ΔXYZ and ΔXTU with T and U lying on XY and XZ respectively.

The value of ∠X = 103°, ∠XUT = 48°, ∠XYZ = 46°

To find: Whether ΔXYZ ∼ ΔXTU or not

We will first find the value of ∠XTU and then compare the corresponding angles of the two triangles

In ΔXTU 103° + 48° + ∠XTU = 180° (Sum of all three angles of triangle is 180°)

∠XTU = 180 − 103 − 48 = 29°

In ΔXTU and ΔXYZ

Only one angle 103° matches hence the AA similarity criterion can not be applied.

hence the two triangles are not similar to each other

ΔXYZ is not similar to the ΔXTU

**Page 363 Exercise 8 Answer**

Given: ∠RST = (3x-9)^{∘} and ∠NSP = (2x+10)^{∘}.

To find the value of x and the value of ∠RST and ∠NSP.

Use the definition of vertically opposite angles, and find the value of x.

Here, ∠RST = ∠NSP [By the definition of vertical opposite angle]

Now to check if the angles RTS and SPN are the same, substitute the value of x in each of the expressions.

∠RTS = x + 19°

=19 + 19

= 38°

∠SPN = 2x

= 2 × 19

= 38°

From this we can see that ∠RTS = ∠SPN

And previously, it was seen that ∠RST = ∠NSP

Hence, AA criterion is satisfied and the two triangles are similar.

For the value of x is 19°, the two triangles are similar, as it satisfies the AA criterion of similarity.

**Page 363 Exercise 9 Answer**

Given: ∠JIH = 43°, ∠JHI = 35° and ∠HFG = 97°

To determine whether △FGH ∼ △JIH.

First, find the value of m ∠J

By using the triangle angle sum theorem, it follows:

m ∠J + 43° + 35° = 180°

m ∠J = 180° − (43° – 35°)

= 102°

By using the vertical angles theorem, it follows:

∠FHG ≅ ∠JHI

By using the definition of congruence, it follows:

m ∠FHG = m ∠JHI

m ∠JHI = 35°

Solve for m ∠G

m ∠G + 35∘ + 97∘ = 180°

m ∠G = 180° − (35° + 97°)

= 48°

Since, all the angles of △FGH and △JIH are not congruent.

Therefore by the definition of similar triangles, it follows the △FGH and △JIH are not similar.

△FGH and △JIH are not similar because the corresponding angles are not congruent.

**Page 363 Exercise 10 Answer**

Given:

To find: Are the given triangles are similar.

Step formulation: First calculate the value of x and then use it to find other angles.

As vertical angles are equal therefore,

4x − 1 = 3x + 14

Take like terms on one side.

4x − 3x = 14 + 1

x = 15

Now find other angles.

∠T = x + 15

= 15 + 15

= 30°

∠P = 2x

= 2 × 15

= 30°

As two angles of the triangles are equal, hence, these are similar triangles.

Yes, the triangles are similar as they have two equal angles.

**Page 363 Exercise 11 Answer**

Given: Two triangles with angles.

To: Describe how to use angle relationships to decide whether any two triangles are similar.

If two angles of one triangle is equal to the two angles of another triangle, then the triangles are known as similar triangles.

This is how angle relationships can be used to check whether any two triangles are similar.

If two angles of one triangle are equal to the two angles of another triangle, then the triangles are known as similar triangles.

This is how angle relationships can be used to check whether any two triangles are similar.

**Page 364 Exercise 12 Answer**

Given:

To: Find whether the given triangles are similar?

Step formulation: Find the angles of both the triangles and then compare.

Sum of interior angles of triangle = 180°

Now, as both the triangles have the same angles, therefore, these are the similar triangles.

Yes, both the triangles have the same angles, therefore, these are the similar triangles.

**Page 364 Exercise 13 Answer**

Given:

To: Find which of the triangles are similar.

Step formulation: First find the angles of the triangles and then compare.

Calculate ∠Z.

X + Y + Z = 180

104 + 45 + Z = 180

Z = 31°

Similarly, ∠H = 45°

∠S = 104°

In ΔXYZ & ΔGHI: ∠Y = ∠H and ∠X = ∠G

Therefore,ΔXYZ ≅ ΔGHI

In ΔSQR & ΔGHI: ∠Q = ∠H and ∠G = ∠S

Therefore,ΔSQR ≅ ΔGHI

In ΔXYZ & ΔSQR: ∠Z = ∠R and ∠Y = ∠Q

Therefore,ΔXYZ ≅ ΔSQR

ΔXYZ ≅ ΔGHI

ΔSQR ≅ ΔGHI

ΔXYZ ≅ ΔSQR

**Page 364 Exercise 14 Answer**

Given:

To: Find whether the given triangles are similar?

Step formulation: First find the angles of the triangles and then compare.

Calculate ∠R.

Q + S + R = 180

59 + 60 + R = 180

R = 61∘

In the triangles ∠R = ∠H and ∠Q = ∠G

Therefore, triangles similar.

Yes, triangles are similar as they have equal angles.