Question

justify the given statement using the picture above.

1. m∠1 + m∠2 = m∠abc
2. if b is the mid - point of $overline{ad}$, then ab = bd.
3. if $overrightarrow{bc}$ bisects ∠fbe, then ∠2≅∠3.
4. if ∠1 and ∠fbd are supplementary, then m∠1 + m∠fbd = 180°
5. if m∠cbd = 90°, then ∠cbd is a right angle.
6. ab + bd = ad
7. if m∠2 = m∠3, then $overrightarrow{bc}$ bisects ∠fbe.
8. if m∠3 + m∠4 = 90°, then ∠3 and ∠4 are complementary.
9. if ∠abc and ∠cbd are right angles, then ∠abc≅∠cbd.
10. if ∠1≅∠2 and ∠2≅∠3, then ∠1≅∠3.

Explanation:

Step1: Angle - addition postulate

The measure of an angle formed by two adjacent angles is the sum of the measures of the two adjacent angles. So, $m\angle1 + m\angle2=m\angle ABC$ because $\angle1$ and $\angle2$ are adjacent and together form $\angle ABC$.

Step2: Definition of mid - point

A mid - point of a line segment divides the line segment into two equal parts. If B is the mid - point of $\overline{AD}$, then by definition $AB = BD$.

Step3: Definition of angle bisector

An angle bisector divides an angle into two congruent angles. If $\overrightarrow{BC}$ bisects $\angle FBE$, then $\angle2\cong\angle3$ by the definition of an angle bisector.

Step4: Definition of supplementary angles

Two angles are supplementary if the sum of their measures is $180^{\circ}$. If $\angle1$ and $\angle FBD$ are supplementary, then $m\angle1 + m\angle FBD=180^{\circ}$ by the definition of supplementary angles.

Step5: Definition of a right angle

An angle with a measure of $90^{\circ}$ is a right angle. If $m\angle CBD = 90^{\circ}$, then $\angle CBD$ is a right angle by the definition of a right angle.

Step6: Segment - addition postulate

The length of a line segment formed by two adjacent line segments is the sum of the lengths of the two adjacent line segments. So, $AB + BD=AD$ because $\overline{AB}$ and $\overline{BD}$ are adjacent segments that form $\overline{AD}$.

Step7: Definition of angle bisector (converse)

If two adjacent angles have equal measures, then the ray between them is an angle bisector. If $m\angle2 = m\angle3$, then $\overrightarrow{BC}$ bisects $\angle FBE$ by the converse of the definition of an angle bisector.

Step8: Definition of complementary angles

Two angles are complementary if the sum of their measures is $90^{\circ}$. If $m\angle3 + m\angle4 = 90^{\circ}$, then $\angle3$ and $\angle4$ are complementary by the definition of complementary angles.

Step9: Definition of congruent angles

All right angles are congruent. If $\angle ABC$ and $\angle CBD$ are right angles, then $\angle ABC\cong\angle CBD$ since the measure of each right angle is $90^{\circ}$.

Step10: Transitive property of congruence

If $\angle1\cong\angle2$ and $\angle2\cong\angle3$, then $\angle1\cong\angle3$ by the transitive property of congruence for angles.

Answer:

1. Angle - addition postulate
2. Definition of mid - point
3. Definition of angle bisector
4. Definition of supplementary angles
5. Definition of a right angle
6. Segment - addition postulate
7. Definition of angle bisector (converse)
8. Definition of complementary angles
9. Definition of congruent angles
10. Transitive property of congruence