Construct a triangle PQR given that PQ = 6 . 2 cm `angle P = 60^(@) and angle Q = 45^(@)` by using ruler and compass only. Measure `angle R `

Answer

To construct triangle PQR with the given conditions (PQ = 6.2 cm, angle P = 60°, and angle Q = 45°), follow these steps:

### Step-by-Step Solution:

1. **Draw Line Segment PQ**:
- Use a ruler to draw a line segment PQ that measures 6.2 cm. Label the endpoints as P and Q.

**Hint**: Ensure that you measure accurately using the ruler for the length of 6.2 cm.

2. **Extend Line Segment PQ**:
- Extend the line segment PQ on both sides to allow space for constructing the angles.

**Hint**: You can use a straightedge or the ruler to extend the line.

3. **Construct Angle P (60°)**:
- Place the compass point on point P and draw a semicircle that intersects the extended line segment PQ.
- Mark the intersection points of the semicircle with the extended line as point A.
- Keeping the compass width the same, place the compass on point A and draw an arc above the line.
- Now, place the compass on point P and draw another arc that intersects the first arc. Label the intersection point as B.
- Draw a straight line from P to B. This line forms angle P = 60° with line segment PQ.

**Hint**: Make sure your compass width remains unchanged when drawing arcs for accurate intersection points.

4. **Construct Angle Q (45°)**:
- Place the compass point on point Q and draw a semicircle that intersects the extended line segment PQ.
- Mark the intersection points of the semicircle with the extended line as point C.
- Keeping the compass width the same, place the compass on point C and draw an arc above the line.
- Now, place the compass on point Q and draw another arc that intersects the first arc. Label the intersection point as D.
- Draw a straight line from Q to D. This line forms a right angle (90°) with line segment PQ.
- To bisect this 90° angle, place the compass on point D and draw an arc. Then, place the compass on point Q and draw another arc that intersects the previous arc. Label the intersection point as E.
- Draw a straight line from Q to E. This line forms angle Q = 45° with line segment PQ.

**Hint**: When bisecting the angle, ensure that the arcs intersect at a point that allows you to draw a straight line accurately.

5. **Label Point R**:
- The intersection of the lines from steps 3 and 4 will be point R.

**Hint**: Ensure that the lines are extended sufficiently to clearly see where they intersect.

6. **Measure Angle R**:
- Use a protractor to measure angle R. Place the protractor's center point at point R and align one side of the angle with the baseline of the protractor. Read the measurement of angle R.

**Hint**: Make sure to read the protractor correctly, checking both the inner and outer scales.

### Conclusion:
After following these steps, you will have constructed triangle PQR with the specified dimensions and angles. The measurement of angle R is found to be 75°.