Question

in exercises a–f, use the two similar triangles.
a. what is ∠l’s corresponding angle?
b. what is ∠m’s corresponding angle?
c. what is ∠n’s corresponding angle?
d. what is side lm’s corresponding side?
e. what is side mn’s corresponding side?
f. what is side ln’s corresponding side?
in exercises g–k, use the two similar trapezoids.
g. what is the measure of ∠t?
h. what is the measure of ∠w?
i. what side corresponds to \\(\overline{pq}\\)?
j. what side corresponds to \\(\overline{rs}\\)?
k. what side corresponds to \\(\overline{qr}\\)?

Explanation:

• For similar triangles, corresponding angles and sides are determined by their position in the similar figures. For similar trapezoids, corresponding angles are equal (as per the properties of similar polygons - corresponding angles of similar polygons are congruent) and corresponding sides are in proportion (but for identification of corresponding sides, we look at the order of vertices in the similarity statement).
• In the triangle part: If \(\triangle LMN\sim\triangle XYZ\) (assuming the order of vertices gives the correspondence), then \(\angle L\) corresponds to \(\angle X\), \(\angle M\) corresponds to \(\angle Z\), \(\angle N\) corresponds to \(\angle Y\), side \(LM\) corresponds to side \(XZ\), side \(MN\) corresponds to side \(ZY\), side \(LN\) corresponds to side \(XY\).
• In the trapezoid part: Since the trapezoids are similar, \(\angle T\) corresponds to \(\angle S\) (so \(m\angle T = 70^{\circ}\)), \(\angle W\) corresponds to \(\angle O\) (so \(m\angle W=110^{\circ}\)). If the trapezoid \(VUTW\sim\) trapezoid \(SPQR\), then side \(VW\) corresponds to side \(PQ\), side \(VU\) corresponds to side \(SP\), side \(UT\) corresponds to side \(QR\), side \(TW\) corresponds to side \(RQ\).

Answer:

A. \(\angle X\)
B. \(\angle Z\)
C. \(\angle Y\)
D. \(XZ\)
E. \(ZY\)
F. \(XY\)
G. \(70^{\circ}\)
H. \(110^{\circ}\)
I. \(VW\)
J. \(VU\)
K. \(UT\)