Question

complete the proof.
given: \\( \angle a d b \\) and \\( \angle b d c \\) form a linear pair and \\( \angle a d b \cong \angle b d c \\).
prove: \\( \angle a d b \\) and \\( \angle b d c \\) are right angles.

Explanation:

Step1: Determine the relationship between ∠ADB and ∠BDC

Since ∠ADB and ∠BDC form a linear pair, by the Linear Pair Theorem, they are supplementary. So, for statement 2, the answer is “supplementary”.

Step2: Find the sum of the measures of ∠ADB and ∠BDC

Supplementary angles add up to \(180^{\circ}\). So, for statement 3, \(m\angle ADB + m\angle BDC=180^{\circ}\), and the reason (for statement 3) is “Definition of supplementary angles”.

Step3: Justify the conclusion for statement 9

A right - angle is an angle whose measure is \(90^{\circ}\). Since \(m\angle ADB = 90^{\circ}\) and \(m\angle BDC=90^{\circ}\), by the definition of a right angle, ∠ADB and ∠BDC are right angles. So, for statement 9, the reason is “Definition of a right angle”.

Answer:

1. supplementary; 3. \(180^{\circ}\), Definition of supplementary angles; 9. Definition of a right angle.