Question

find each measure.
m∠cad =
type your answer...
m∠acd =
type your answer...
m∠acb =
type your answer...

Explanation:

Step1: Analyze triangle \( ACD \)

In \( \triangle ACD \), \( AD = CD \) (marked with equal segments) and \( \angle D = 90^\circ \), so \( \triangle ACD \) is an isosceles right triangle.
For an isosceles right triangle, the two acute angles are equal. Let \( m\angle CAD = m\angle ACD = x \).
Using the angle - sum property of a triangle (\( \angle D+\angle CAD+\angle ACD = 180^\circ \)):
\( 90^\circ+x + x=180^\circ \)
\( 2x=180^\circ - 90^\circ=90^\circ \)
\( x = 45^\circ \). So \( m\angle CAD = 45^\circ \) and \( m\angle ACD = 45^\circ \).

Step2: Find \( m\angle ACB \)

\( \angle ACD \) and \( \angle ACB \) are supplementary (they form a linear pair), so \( m\angle ACB=180^\circ - m\angle ACD \)
Since \( m\angle ACD = 45^\circ \), then \( m\angle ACB = 180^\circ-45^\circ = 135^\circ \)

Answer:

s:
\( m\angle CAD=\boldsymbol{45^\circ} \)
\( m\angle ACD=\boldsymbol{45^\circ} \)
\( m\angle ACB=\boldsymbol{135^\circ} \)