**Problem 1 :**

In the diagram shown below

PQ ∥ ST, QS = 8, SR = 4 and PT = 12

Find the length of TR.

**Problem 2 :**

In the diagram shown below KL ∥ MN, find the values of the x and y.

**Problem 3 : **

In the diagram shown below, determine whether MN ∥ GH.

**Problem 4 :**

In the diagram shown below,

∠1 ≅ ∠2 ≅ ∠3

PQ = 9, QR = 15 and ST = 11

Find the length of TU.

**Problem 5 :**

In the diagram shown below, ∠CAD ≅ ∠DAB. Use the given side lengths to find the length of DC.

**Problem 6 :**

We are insulating your attic, as shown. The vertical 2 x 4 studs are evenly spaced. Explain why the diagonal cuts at the tops of the strips of insulation should have the same lengths.

**Problem 1 :**

In the diagram shown below

PQ ∥ ST, QS = 8, SR = 4 and PT = 12

Find the length of TR.

**Solution : **

By Triangle Proportionality Theorem,

SR / QS = TR / PT

Substitute.

4 / 8 = TR / 12

Simplify.

1 / 2 = TR / 12

Multiply each side by 12.

12 ⋅ (1 / 2) = (TR / 12) ⋅ 12

6 = TR

So, the length of TR is 6 units.

**Problem 2 :**

In the diagram shown below KL ∥ MN, find the values of the x and y.

**Solution : **

**Finding the value of x :**

To find the value of x, we can set up a proportion.

Write proportion.

9 / 13.5 = (37.5 - x) / x

By cross product property of proportion,

9x = 13.5(37.5 - x)

9x = 506.25 - 13.5x

Add 13.5x to each side.

22.5x = 506.25

Divide each side by 22.5

x = 22.5

**Finding the value of y :**

Since KL ∥ MN and ΔJKL ∼ ΔJMN,

JK / JM = KL / MN

JK / (JK + KM) = KL / MN

9 (9 + 13.5) = 7.5 / y

9 / 22.5 = 7.5 / y

By cross product property of proportion,

9y = 7.5 ⋅ 22.5

9y = 168.75

Divide each side by 9.

y = 18.75

**Problem 3 : **

In the diagram shown below, determine whether MN ∥ GH.

**Solution :**

Begin by finding and simplifying the ratios of the two sides divided by MN.

LM / MG = 56 / 21 = 8 / 3

LN / NH = 48 / 16 = 3 / 1

Because 8 / 3 ≠ 3 / 1, MN is not parallel to GH.

**Problem 4 :**

In the diagram shown below,

∠1 ≅ ∠2 ≅ ∠3

PQ = 9, QR = 15 and ST = 11

Find the length of TU.

**Solution : **

Because corresponding angles are congruent the lines are parallel and we can use Theorem 1 on Proportionality.

Parallel lines divide transversals proportionally.

PQ / QR = ST / TU

Substitute.

9 / 15 = 11 / TU

Simplify.

3 / 5 = 11 / TU

By reciprocal property of proportion,

5 / 3 = TU / 11

Multiply each side by 11.

11 ⋅ (5 / 3) = (TU / 11) ⋅ 11

55 / 3 = TU

Hence, the length TU is 55 / 3 or 18⅓ units.

**Problem 5 :**

In the diagram shown below, ∠CAD ≅ ∠DAB. Use the given side lengths to find the length of DC.

**Solution : **

Since AD is an angle bisector of ∠CAB, we can apply Theorem 2 on Proportionality.

Let x = DC.

Then,

BD = 14 - x

Apply Theorem 2 on Proportionality.

AB / AC = BD / DC

Substitute.

9 / 15 = (14 - x) / x

3 / 5 = (14 - x) / x

Multiply each side by 5x.

5x ⋅ (3 / 5) = [(14 - x) / x] ⋅ 5x

3x = 70 - 5x

Add 5x to each side.

8x = 70

Divide each side by 8.

x = 8.75

So, the length of DC is 8.75 units.

**Problem 6 :**

We are insulating your attic, as shown. The vertical 2 x 4 studs are evenly spaced. Explain why the diagonal cuts at the tops of the strips of insulation should have the same lengths.

**Solution : **

Because the studs AD, BE and CF are each vertical, we know that they are parallel to each other. Using Theorem 8.6, you can conclude that

DE / EF = AB / BC

Because the studs are evenly spaced, we know that

DE = EF

So, we can conclude that

AB = BC

which means that the diagonal cuts at the tops of the strips have the same lengths.

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