Question

match each sentence with the correct set-up. not all set-ups will be used. if ∠a and ∠b are vertical angles, then __. if d is in the interior of ∠abc, then . if m is the midpoint of ab, then . if ∠a and ∠b form a linear pair, then . if \\(\overrightarrow{ad}\\) bisects ∠bac, then __. drag & drop the answer: \\(m\angle a + m\angle b = 180^\circ\\), \\(m\angle a + m\angle b = 90^\circ\\), \\(\angle a \cong \angle b\\), \\(m\angle a = m\angle b\\), \\(\angle bad \cong \angle dac\\), \\(m\angle bad = m\angle dac\\), \\(m\angle abd + m\angle dbc = m\angle abc\\), \\(am \cong mb\\), \\(am = mb\\), \\(m\angle abc + m\angle bcd = m\angle abd\\), \\(ab \cong bc\\), \\(ab = bc\\)

Explanation:

Step1: Analyze "If ∠A and ∠B are vertical angles"

Vertical angles are equal, so \( m\angle A = m\angle B \) and \( \angle A \cong \angle B \). So match with \( \angle A \cong \angle B, m\angle A = m\angle B \).

Step2: Analyze "If D is in the interior of ∠ABC"

By the Angle Addition Postulate, \( m\angle ABD + m\angle DBC = m\angle ABC \). So match with \( m\angle ABD + m\angle DBC = m\angle ABC \).

Step3: Analyze "If M is the midpoint of \( \overline{AB} \)"

Midpoint divides the segment into two congruent and equal parts, so \( \overline{AM} \cong \overline{MB}, AM = MB \). So match with \( \overline{AM} \cong \overline{MB}, AM = MB \).

Step4: Analyze "If ∠A and ∠B form a linear pair"

Linear pair angles sum to \( 180^\circ \), so \( m\angle A + m\angle B = 180^\circ \). So match with \( m\angle A + m\angle B = 180^\circ \).

Step5: Analyze "If \( \overrightarrow{AD} \) bisects \( \angle BAC \)"

Angle bisector divides the angle into two congruent and equal angles, so \( \angle BAD \cong \angle DAC, m\angle BAD = m\angle DAC \). So match with \( \angle BAD \cong \angle DAC, m\angle BAD = m\angle DAC \).

Answer:

• If \( \angle A \) and \( \angle B \) are vertical angles, then \( \boldsymbol{\angle A \cong \angle B, m\angle A = m\angle B} \).
• If \( D \) is in the interior of \( \angle ABC \), then \( \boldsymbol{m\angle ABD + m\angle DBC = m\angle ABC} \).
• If \( M \) is the midpoint of \( \overline{AB} \), then \( \boldsymbol{\overline{AM} \cong \overline{MB}, AM = MB} \).
• If \( \angle A \) and \( \angle B \) form a linear pair, then \( \boldsymbol{m\angle A + m\angle B = 180^\circ} \).
• If \( \overrightarrow{AD} \) bisects \( \angle BAC \), then \( \boldsymbol{\angle BAD \cong \angle DAC, m\angle BAD = m\angle DAC} \).