Exercise 84 Page 578

Practice makes perfect

a Examining the diagram, we can see two parallel lines that are cut by a transversal. The labeled angles, x and y, make a pair of alternate interior angles.

Since the two lines are parallel, x and y must be congruent by the Alternate Interior Angles Theorem. x=y

b If we assume that y is the circle's central angle, then the inscribed angle will be half the size of the central angle.

c The chords forms two pairs of corresponding sides in similar triangles.

Therefore, we can write the following equation. y/x=5/3 ⇔ 3y=5x

d The angles, x and y, are the inscribed angles of two different central angles.

Since ∠ a and ∠ b sum to 360^(∘) and ∠ x and ∠ y are half the size of a and b, respectively, the sum of ∠ x and ∠ y must equal 180^(∘). m∠ x+m∠ y=180^(∘)