Exercise 42 Page 225

Practice makes perfect

a In the diagram, we can make out two triangles. If the triangles are similar, they have at least two pairs of congruent angles. First, we notice that they share an angle.

We can also identify a pair of corresponding angles and since the two sides cut by the third side are parallel, we know by the Corresponding Angles Theorem that they are congruent.

Since two pairs of angles are congruent, we can claim that the triangles are similar by the AA Similarity condition. Knowing the triangles similar, we can now write and solve an equation for x.

x+13/13=13/8

▼

Solve for x

MultEqn

LHS * 13=RHS* 13

x+13=13/8* 13

MoveRightFacToNum

a/c* b = a* b/c

x+13=169/8

UseCalc

Use a calculator

x+13=21.125

SubEqn

LHS-13=RHS-13

x=8.125

b From the diagram, we can make out one pair of corresponding angles. Since the two sides cut by the third side are parallel, we know by the Corresponding Angles Theorem that they are congruent.

However, we do not know the measure of any other angle and therefore, we cannot claim similarity.

c To prove that the triangles are similar, we have to show that at least two pairs of angles in the triangles are congruent. Examining the diagram, we can immediately identify a pair of vertical angles. By the Vertical Angles Theorem, we know that these are congruent.

We can also identify a pair of alternate interior angles. Since the two sides cut by the third side are parallel, we know by the Alternate Interior Angles Theorem that they are congruent.

Now, we know that the two triangles have two pairs of congruent angles and, therefore, the triangles are similar by the AA Similarity condition. In similar triangles, the ratio of corresponding sides is always equal. Therefore, by identifying corresponding sides in our triangles, we can write an equation that involves x.