Question

1. m∠7 and m∠8 are complementary, ∠5 ≅ ∠8 and m∠6 = 29° m∠5 = ______ m∠6 = ____ m∠7 = ____ m∠8 = ______

Explanation:

Step1: Determine \(m\angle5\)

Since \(\angle5\cong\angle8\), their measures are equal. Given \(m\angle6 = 29^{\circ}\), but from the congruence \(m\angle5=m\angle8\). However, we know that \(\angle7\) and \(\angle8\) are complementary (\(m\angle7 + m\angle8=90^{\circ}\)) and also from the straight - line (sum of angles \(\angle5+\angle6+\angle7+\angle8 = 180^{\circ}\)). But since \(\angle5\cong\angle8\), let \(m\angle5 = x\) and \(m\angle8=x\).
We know \(m\angle6 = 29^{\circ}\). And because \(\angle7\) and \(\angle8\) are complementary \(m\angle7=90 - x\).
Substituting into \(\angle5+\angle6+\angle7+\angle8 = 180^{\circ}\), we get \(x + 29+(90 - x)+x=180\). Simplifying gives \(x+119 = 180\), so \(x = 61^{\circ}\). But wait, another approach: Since \(\angle5\cong\angle8\), and \(\angle7\) and \(\angle8\) are complementary. Also, from the straight - line \(\angle5+\angle6+\angle7+\angle8 = 180^{\circ}\). Since \(\angle5=\angle8\), we can rewrite it as \(2\angle5+\angle6+\angle7 = 180^{\circ}\). But \(\angle7 = 90^{\circ}-\angle8=90^{\circ}-\angle5\). Substitute \(\angle7\) into the equation: \(2\angle5+\angle6+(90^{\circ}-\angle5)=180^{\circ}\). Then \(\angle5+\angle6 + 90^{\circ}=180^{\circ}\). Given \(\angle6 = 29^{\circ}\), so \(m\angle5=180^{\circ}-90^{\circ}-29^{\circ}=61^{\circ}\)

Step2: Determine \(m\angle6\)

Given directly \(m\angle6 = 29^{\circ}\)

Step3: Determine \(m\angle7\)

Since \(\angle7\) and \(\angle8\) are complementary (\(m\angle7+m\angle8 = 90^{\circ}\)) and \(m\angle8=m\angle5 = 61^{\circ}\), then \(m\angle7=90^{\circ}-m\angle8\). Substitute \(m\angle8 = 61^{\circ}\), so \(m\angle7=90 - 61=29^{\circ}\)

Step4: Determine \(m\angle8\)

Since \(\angle5\cong\angle8\), \(m\angle8=m\angle5 = 61^{\circ}\)

Answer:

\(m\angle5 = 61^{\circ}\)
\(m\angle6 = 29^{\circ}\)
\(m\angle7 = 29^{\circ}\)
\(m\angle8 = 61^{\circ}\)