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ABCD is a parallelogram - Edexcel - GCSE Maths - Question 21 - 2018 - Paper 3
Question 21
ABCD is a parallelogram. ABP and QDC are straight lines. Angle ADP = angle CBQ = 90° (a) Prove that triangle ADP is congruent to triangle CBQ. (b) Explain why AQ i... show full transcript
Worked Solution & Example Answer:ABCD is a parallelogram - Edexcel - GCSE Maths - Question 21 - 2018 - Paper 3
Step 1
Prove that triangle ADP is congruent to triangle CBQ.
Answer
To prove that triangles ADP and CBQ are congruent, we will use the Angle-Side-Angle (ASA) postulate.
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Identify Equal Angles:
- We know from the given information that angle ADP is equal to angle CBQ, as both are 90°.
- Additionally, since ABCD is a parallelogram, opposite angles of a parallelogram are equal. Thus, angle DAP = angle BQC.
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Identify Equal Sides:
- The sides AD and CB are corresponding sides of the parallelogram ABCD, so AD = CB.
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Conclusion of Congruency:
- We now have two angles and one side from the two triangles equal:
- Angle ADP = Angle CBQ (90°)
- Angle DAP = Angle BQC
- Side AD = Side CB
- Therefore, by the ASA criterion, triangle ADP is congruent to triangle CBQ.
- We now have two angles and one side from the two triangles equal:
Step 2
Explain why AQ is parallel to PC.
Answer
In parallelogram ABCD, opposite sides are parallel by definition. Thus, AD is parallel to BC.
Furthermore, since ABP and QDC are straight lines intersecting each other at points A and C, it follows that:
- Identify Parallel Sides:
- AD is parallel to BC (opposite sides of the parallelogram).
- By corresponding angles, the angles formed at A and C where AQ and PC are extended would equal 180° (because they are co-interior angles).
Thus, AQ is parallel to PC.