Question

prove that \\( \angle abc \\) is a right angle.

given: \\( \overline{ad} \cong \overline{bd} \cong \overline{cd} \\)

prove: \\( m\angle abc = 90 \\)

\\( \overline{ad} \cong \overline{bd} \cong \overline{cd} \\) given
\\( \triangle adb \\) and \\( \triangle cdb \\) are isosceles. definition of isosceles triangles
\\( m\angle dab = x \\) given
\\( m\angle dba = x \\) isosceles triangle theorem
\\( m\angle cdb = 2x \\) triangle exterior angle theorem
\\( m\angle bcd + m\angle dbc + m\angle cdb = 180 \\) triangle angle - sum theorem
\\( m\angle bcd = m\angle dbc \\) isosceles triangle theorem
\\( 2\cdot m\angle bcd + 2x = 180 \\) substitution
\\( m\angle bcd = \square \\) (do not include the degree symbol in your answer.) algebraic simplification

Explanation:

Step1: Solve the equation \(2\cdot m\angle BCD + 2x=180\) for \(m\angle BCD\)

First, subtract \(2x\) from both sides of the equation:
\(2\cdot m\angle BCD=180 - 2x\)

Step2: Divide both sides by 2

\(m\angle BCD=\frac{180 - 2x}{2}\)
Using the distributive property \(\frac{a - b}{c}=\frac{a}{c}-\frac{b}{c}\) (where \(a = 180\), \(b=2x\), \(c = 2\)), we get \(m\angle BCD = 90 - x\)

Answer:

\(90 - x\)