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ABCD is a parallelogram - Edexcel - GCSE Maths - Question 1 - 2018 - Paper 1
Question 1
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 1 - 2018 - Paper 1
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) criterion.
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Identify equal angles:
- Since ABCD is a parallelogram, we know opposite angles are equal, so:
- Angle DAB = Angle ABC
- Given that Angle ADP = Angle CBQ = 90°, we can confirm two angles in each triangle are equal.
- Since ABCD is a parallelogram, we know opposite angles are equal, so:
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Identify equal sides:
- Both triangles share side DP = CB, as they are corresponding segments in the parallel lines.
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Establish congruence:
- We have identified two angles and the included side in both triangles:
- Angle ADP = Angle CBQ
- Angle DAP = Angle CBA (because they are alternate angles)
- Side DP = Side CB
- We have identified two angles and the included side in both triangles:
Thus, by the ASA criterion, triangle ADP is congruent to triangle CBQ.
Step 2
Explain why AQ is parallel to PC.
Answer
To explain why line segment AQ is parallel to line segment PC, we refer to the properties of a parallelogram. In parallelograms, the opposite sides are parallel. Since ABCD is given as a parallelogram, we can conclude:
- Sides AD and BC are parallel.
- From the definitions, line AQ extends parallel to line segment PC due to the nature of opposite sides in a parallelogram.
Therefore, we can confidently state that AQ is parallel to PC.