Question

Complete the proof $m\angle 2>m\angle 4$if $m\angle 2>m\angle 1$is given.

Short answer

Statement	Reason
a.$m\angle 2>m\angle 1$	Given
b.$m\angle 1>m\angle 3$	Exterior angle is greater than any interior angle of the triangle
c.$m\angle 2>m\angle 3$	From a.
d.$\angle 3\cong \angle 4$, or$m\angle 3=m\angle 4$	Vertically opposite angles
e.$m\angle 2>m\angle 4$	From c.

Step-by-step solution

Step 1. Point the lines in the given figure

Step 2. Define vertically opposite angles and interior, exterior angles

Pair of opposite angles formed by intersecting lines.

Therefore, angles 3 and 4 are vertically opposite angles. Angle 1 and 4 are exterior angles of the triangle APQ and 3 is an interior angle.

Step 3. Prove that angle 2 is greater than angle 4

Proof:

$m\angle 2>m\angle 1$ …… (1) (Given)

$m\angle 1>m\angle 3$ (Exterior angle is greater than any interior angle of the triangle)

$m\angle 2>m\angle 3$ …… (2) (From equation (1))

$m\angle 4=m\angle 3$ (Vertically opposite angles)

$m\angle 2>m\angle 4$ (From equation (2))

Hence proved $m\angle 2>m\angle 4$.