The third part of this series about congruent figures is proving triangles congruent. At this point, there is only one way to prove triangles congruent, with the definition of congruent triangles. Congruent triangles have 3 pairs of corresponding angles and 3 pairs of congruent angles. So, to prove two triangles congruent, we must establish that all 3 pairs of corresponding sides and angles are congruent.

Given: segment WX is congruent to YX, segment WZ is congruent to segment YZ, angle W is congruent to angle Y and segment XZ is perpendicular to segment WY

Prove: triangle WXZ is congruent to triangle YXZ

With the given information: segment WX is congruent to YX, segment WZ is congruent to segment YZ, angle W is congruent to angle Y and segment XZ is perpendicular to segment WY, prove triangle WXZ is congruent to triangle YXZ. First, segment XZ is congruent to segment XZ by the reflexive property of congruence. Angles XZW and XZY are right angles by the definition of right angles. Angle XZW and angle XZY are congruent because all right angles are congruent. Angle WXZ is congruent to angle YXZ by the third angle theorem. It can be established that triangle WXZ is congruent to triangle YXZ by the definition of congruent triangles.

The second part of this series about congruent figures is proving triangles congruent. If you are interested in the reading the first post on Congruent Figures. At this point, there is only one way to prove triangles congruent, with the definition of congruent triangles. Congruent triangles have 3 pairs of corresponding angles and 3 pairs of congruent angles. So, to prove two triangles congruent, we must establish that all 3 pairs of corresponding sides and angles are congruent. The third angle theorem states, if two angles of one triangle are congruent to two angles of another triangle, the third angles are congruent to each other.

Given: segment AB is congruent to segment AD, segment BC is congruent to segment DC, angle B is congruent to angle D, and angle BAC is congruent angle DAC

Prove: triangle ABC is congruent to triangle ADC

Prove Triangle ABC congruent to Triangle ADC

With the given information: segment AB is congruent to segment AD, segment BC is congruent to segment DC, angle B is congruent to angle D, and angle BAC is congruent angle DAC, prove triangle ABC is congruent to triangle ADC. Segment AC is congruent to segment AC by the reflexive property of congruence. Next, it must be established that angle ACB is congruent to angle ACD, by the third angle theorem. Since, all three pairs of corresponding sides and angles are congruent, triangle ABC is congruent to triangle ADC.