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Write a proof that line AC is congruent to BD.
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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.
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)
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