Question

part 2 - missing side lengths & angles measures

1. the two triangles are congruent. find all missing sides and angles of each triangle.

Explanation:

Step1: Find missing angles in each triangle

In right - triangle \(RPK\), one angle is \(90^{\circ}\) and another is \(55^{\circ}\). Using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)), the third angle \(\angle K = 180^{\circ}-(90^{\circ}+55^{\circ}) = 35^{\circ}\). In right - triangle \(FHA\), one angle is \(90^{\circ}\) and another is \(35^{\circ}\), so the third angle \(\angle A=180^{\circ}-(90^{\circ}+35^{\circ}) = 55^{\circ}\). Since the two triangles are congruent, corresponding angles are equal.

Step2: Find missing sides using congruence

Corresponding sides of congruent triangles are equal. If we assume that the side of length \(9\) in \(\triangle RPK\) corresponds to a side in \(\triangle FHA\) and the side of length \(11\) in \(\triangle FHA\) corresponds to a side in \(\triangle RPK\). Let's say \(RP\) corresponds to \(FH\), \(PK\) corresponds to \(HA\), and \(RK\) corresponds to \(FA\). So, if \(PK = 9\), then \(HA = 9\); if \(FH = 11\), then \(RP = 11\). To find \(RK\) and \(FA\) (the hypotenuses), we use the Pythagorean theorem. In \(\triangle RPK\), \(RK=\sqrt{RP^{2}+PK^{2}}=\sqrt{11^{2}+9^{2}}=\sqrt{121 + 81}=\sqrt{202}\). In \(\triangle FHA\), \(FA=\sqrt{FH^{2}+HA^{2}}=\sqrt{11^{2}+9^{2}}=\sqrt{202}\).

Answer:

In \(\triangle RPK\): \(\angle K = 35^{\circ}\), \(RP = 11\), \(RK=\sqrt{202}\)
In \(\triangle FHA\): \(\angle A = 55^{\circ}\), \(HA = 9\), \(FA=\sqrt{202}\)