Construct the triangle ABC, where |AB| = 10 cm, |\angle AB| = 60° and |\angle ABC| = 40°. Label each vertex clearly. Measure |BC|, and write your answer in cm, correct to 1 decimal place. The diagram shows a parallelogram with vertices P, Q, R, and S. |\angle SPQ| = 115°, |\angle QRS| = α° and |\angle RSP| = β°. (i) Write down the value of α and the value of β. (ii) Explain why the triangle PQR is congruent to triangle RSP. Give a reason for any statement you make in your explanation.

Leaving Cert Mathematics Geometry - Proofs & Constructions: Construct the triangle ABC, wher

Construct the triangle ABC, where |AB| = 10 cm, |\angle AB| = 60° and |\angle ABC| = 40° - Leaving Cert Mathematics - Question 6 - 2018

Question 6

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Construct the triangle ABC, where |AB| = 10 cm, |\angle AB| = 60° and |\angle ABC| = 40°. Label each vertex clearly. Measure |BC|, and write your answer in cm, corr... show full transcript

Worked Solution & Example Answer:Construct the triangle ABC, where |AB| = 10 cm, |\angle AB| = 60° and |\angle ABC| = 40° - Leaving Cert Mathematics - Question 6 - 2018

Step 1

Construct the triangle ABC

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Answer

To draw triangle ABC:

Start by drawing line segment AB measuring 10 cm.
At point A, use a protractor to measure an angle of 60° and draw a ray towards the left. This will start forming angle CAB.
At point B, also use the protractor to measure an angle of 40° and draw a ray towards the right. This will start forming angle ABC.
The point where these two rays intersect is point C. Label the vertices clearly.

Step 2

Measure |BC|, and write your answer in cm, correct to 1 decimal place.

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Answer

|BC| can be measured using a ruler. After measuring, if |BC| equals 8 cm, write the final answer as:

|BC| = 8.0 cm

Step 3

Write down the value of α and the value of β.

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Answer

Given that |\angle SPQ| = 115°,

Since opposite angles in a parallelogram are equal, |\angle QRS| (which is α) will be:
- α = 115°.

To find β:

The consecutive angles in a parallelogram are supplementary, hence:
- β = 180° - 115° = 65°.

Step 4

Explain why the triangle PQR is congruent to triangle RSP.

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Answer

The triangles PQR and RSP are congruent by the criteria SSS (Side-Side-Side) because:

|PQ| = |SR| (Opposite sides of the parallelogram are equal).
|QR| = |PS| (Also, opposite sides of the parallelogram are equal).
The angle |\angle PQR| = |\angle RSP| = 65° (As they are alternate interior angles due to the parallel sides PS and QR)

Thus, by SSS criteria, triangle PQR is congruent to triangle RSP.

Construct the triangle ABC, where |AB| = 10 cm, |\angle AB| = 60° and |\angle ABC| = 40° - Leaving Cert Mathematics - Question 6 - 2018

Worked Solution & Example Answer:Construct the triangle ABC, where |AB| = 10 cm, |\angle AB| = 60° and |\angle ABC| = 40° - Leaving Cert Mathematics - Question 6 - 2018

Construct the triangle ABC

Measure |BC|, and write your answer in cm, correct to 1 decimal place.

Write down the value of α and the value of β.

Explain why the triangle PQR is congruent to triangle RSP.

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