Exercise 21 Page 566

Let's review the theorems that can help us prove that two triangles are similar.

1. Angle-Angle Similarity Theorem. If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
2. Side-Side-Side Similarity Theorem. If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
3. Side-Angle-Side Similarity Theorem. If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

Now we will identify the similar triangles and find the measures, one at a time.

Similar Triangles

Notice that ∠ GHJ is congruent to ∠ GDH. We can also see that △ GHJ and △ GDH share ∠ G. This means that two angles of △ GHJ are congruent to two angles of △ GDH. Therefore, by the Angle-Angle Similarity Theorem, △ GHJ and △ GDH are similar. △ GHJ ~ △ GDH

Finding the Measures

Using our similarity statement from above, we can identify two pairs of corresponding sides that will help us find the requested lengths. DH corresponds with HJ GD corresponds with GH Recall that corresponding segments of similar figures will have proportional lengths. We are given expressions for the lengths of these sides which we can use to write a proportion. DH/HJ = GD/GH ⇓ 2x+4/10= 2x-2/7 Let's solve this equation to find x.

2x+4/10= 2x-2/7

▼

Solve for x

CrossMult

Cross multiply

7(2x+4)=10(2x-2)

Distr

Distribute 7

14x+28=10(2x-2)

Distr

Distribute 10

14x+28=20x-20

SubEqn

LHS-28=RHS-28

14x=20x-48

SubEqn

LHS-20x=RHS-20x

-6x=-48

DivEqn

.LHS /(-6).=.RHS /(-6).

x=8

Now that we know the value of x, we can find GD and DH. We will substitute x=8 in the expressions for the lengths.

Measure	Expression	x=8	Simplified
GD	2x-2	2(8)-2	14
DH	2x+4	2(8)+4	20

We found that GD=14 and DH=20.