Chapter 4.2, Problem 24WE

To determine

To draw: and label a diagram, and to prove the given statement.

Explanation of Solution

Given information:

Given statement:

A line perpendicular to $\overline{AB}$ passes through the midpoint $\overline{AB}$ , and segments are drawn from any other point on that line to $A$ and $B$ , then two congruent triangles are formed.

Calculation:

SSS Postulate: if there sides of triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.

SAS Postulate: If two sides and the angle included between them of triangle are equal to the two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.

ASA Postulate: If two angles and the included side of triangle are equal to the two corresponding angles and the included sides of another triangle, then the triangles are congruent.

Given:

Line $\overline{CD}$ is perpendicular to $\overline{AB}$ ;

$D$ is the midpoint of $\overline{AB}$

Prove: $\Delta ADC\cong \Delta ADB$

Proof:

Statements	Reasons
1. $D$ is the midpoint of   $\overline{AB}$	Given
2. $\overline{AD}\cong \overline{BD}$	Definition of midpoint
3. Line   $\overline{CD}$ is perpendicular to   $\overline{AB}$	Given
4. $m\angle CDA=90°$ ;   $m\angle CDB=90°$	Definition of perpendicular lines
5. $\angle CDA\cong \angle CDB$	Definition of congruent angles
6. $\overline{CD}\cong \overline{CD}$	Reflexive property
7. $\Delta ADC\cong \Delta ADB$	SAS Postulate