# I really need help on this question that i'm stuck on. I'll really appreciate it! Write a proof that line AC is congruent to BD.             ...

I really need help on this question that i'm stuck on. I'll really appreciate it!

Write a proof that line AC is congruent to BD.

Statements                                Reasons

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Concepts Alternate Interior Angles Converse, Reflexive Property, Parallelograms or Triangle Congruency.

Remember to state your givens and definitions, you need to build up the step-by-step logical case that AC is congruent to BD.

Solution using Parallelograms

angles CBA and DAB are congruent - Given

angles CAB and DBA are right angles (and congruent) - Given

angles CBA and DAB are alternate interior angles - Definition

segments BC and AD are parallel - alternate interior angles converse

angles CAB and DBA are alternate interior angles - Definition

segments AC and BD are parallel - alternate interior angles

(ACBD is a quadrilateral - definition)

ACBD is a parallelogram - Parallelogram Converse

AC is congruent to BD - ACBD is a parallelogram, opposite sides are congruent.

ALTERNATE solution - using ASA Triangle Congruency

Once you have identified congruence of the angles (see above)...

AB is congruent to BA - reflexive property

Triangles ABD and BAC are congruent - Angle-Side-Angle Congruency

AC is congruent to BD - CPCTC: Corresponding parts of congruent triangles are congruent.

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In triangle CAB    and   triangle DBA

angle CAB   =   angle DBA          (given)

AB   =   BA                  (same line )

angle ABC  =    angle BAD        (given)

There fore     triangle CAB congruence to triangle DBA    by ASA .

Thus       AC=BD     ( congruent part of congruent triangle)