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ABCD is a parallelogram - Edexcel - GCSE Maths - Question 5 - 2018 - Paper 3
Question 5
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 5 - 2018 - Paper 3
Step 1
Prove that triangle ADP is congruent to triangle CBQ.
Answer
To prove that triangle ADP is congruent to triangle CBQ, we can use the Angle-Side-Angle (ASA) congruence criterion.
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Identify Equal Angles: From the information provided, we know that angle ADP and angle CBQ are both right angles (90°).
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Identify Equal Sides: Since ABCD is a parallelogram, opposite sides are equal. Thus, we have:
- AD = BC (opposite sides of a parallelogram)
- DP = BQ (both are segments connecting corresponding vertices)
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Identify the Third Angle: Since both triangles share angle DAP = angle CBQ, we can now state:
- Thus, we have established two pairs of equal angles (ADP = CBQ = 90° and DAP = CBQ) and a pair of equal sides (AD = BC).
By the ASA criterion, triangle ADP is congruent to triangle CBQ.
Step 2
Explain why AQ is parallel to PC.
Answer
In a parallelogram, the opposite sides are always parallel.
Since ABCD is a parallelogram, sides AB and DC are parallel. Since AQ and PC both extend from the same line segments of a parallelogram and do not intersect due to the properties of parallel lines, we can conclude that:
- AQ is parallel to PC because they are corresponding sides of the segments formed by extending sides of the parallelogram, adhering to the property that opposite sides of a parallelogram are parallel.