Question

collaborative activity: complementary angles

1. right angles drk and cxw are shown. ray rl is inside ∠drk, xh is inside ∠cxw, ( mangle drl = 28^circ ), and ( mangle cxh = 28^circ ).

a. use the definition of complementary angles found in the lesson summary to explain why ( angle lrd ) and ( angle lrk ) are complementary angles.

b. determine the measure of ( angle lrk ).

Explanation:

Part a

Step1: Recall complementary angles definition

Complementary angles sum to \(90^\circ\).

Step2: Analyze \(\angle DRK\)

\(\angle DRK\) is a right angle, so \(m\angle DRK = 90^\circ\).

Step3: Express \(\angle DRK\) as sum

\(\angle DRK=\angle DRL + \angle LRK\) (angle addition postulate).

Step4: Substitute and conclude

Given \(m\angle DRL = 28^\circ\), then \(m\angle DRL + m\angle LRK=90^\circ\). By complementary angles definition, \(\angle DRL\) (same as \(\angle LRD\)) and \(\angle LRK\) are complementary.

Part b

Step1: Use angle sum for \(\angle DRK\)

Since \(\angle DRK = 90^\circ\) and \(\angle DRK=\angle DRL+\angle LRK\), we have \(m\angle DRL + m\angle LRK = 90^\circ\).

Step2: Substitute known measure

Given \(m\angle DRL = 28^\circ\), substitute: \(28^\circ + m\angle LRK = 90^\circ\).

Step3: Solve for \(m\angle LRK\)

Subtract \(28^\circ\) from both sides: \(m\angle LRK = 90^\circ - 28^\circ = 62^\circ\).

Answer:

s:
a. \(\angle LRD\) and \(\angle LRK\) are complementary because their measures sum to \(90^\circ\) (as \(\angle DRK = 90^\circ=\angle LRD+\angle LRK\)).
b. \(m\angle LRK = \boldsymbol{62^\circ}\)