Geometry and proof · Ages 13–14
Bisecting an angle and a line segment worksheet
Part of: Congruent triangles
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Bisecting an angle and a line segment worksheet
Geometry and proof · Ages 13–14
Numsprite
numsprite.com
Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.Bisect ∠AOB (98°) with compasses and a straight edge: put the steps in order.
- A. ∠AOR = ∠BOR = 49°: join OR
- B. PR = QR: with centres P and Q and one radius, draw arcs crossing at R
- C. OP = OQ: with centre O, draw an arc cutting OA at P and OB at Q
Write the letters in the right order
Answer:
2.Draw the perpendicular bisector of the line segment AB (8 cm) with compasses and a straight edge: put the steps in order.
- A. AQ = BQ: with the same radius, draw arcs again below AB, crossing at Q
- B. AP = BP: with centres A and B and one radius (more than 4 cm), draw arcs crossing above AB at P
- C. AM = MB = 4 cm: join PQ, cutting AB at M
Write the letters in the right order
Answer:
3.PQ is the perpendicular bisector of AB, cutting it at M, as shown. AB = 10 cm, AM =
Answer:
4.Bisect ∠AOB (70°) with compasses and a straight edge: put the steps in order.
- A. ∠AOR = ∠BOR = 35°: join OR
- B. OP = OQ: with centre O, draw an arc cutting OA at P and OB at Q
- C. PR = QR: with centres P and Q and one radius, draw arcs crossing at R
Write the letters in the right order
Answer:
5.P is on the perpendicular bisector of the line segment AB, PA = 9 cm and AB = 8 cm. PB =
Answer:
6.OR bisects ∠AOB, as shown, and ∠AOB = 122°. ∠AOR =
Answer:
7.P is on the perpendicular bisector of the line segment AB, PA = 7 cm and AB = 2 cm. PB =
Answer:
8.OR bisects ∠AOB, as shown, and ∠AOB = 124°. ∠AOR =
Answer:
9.The perpendicular bisector PQ of AB cuts it at M, as shown; AB = 10 cm and PA = 11 cm. PB = cm, MB = cm, ∠PMB = °.
Fill in each blank
Answer:
10.The perpendicular bisector PQ of AB cuts it at M, as shown; AB = 10 cm and PA = 7 cm. PB = cm, MB = cm, ∠PMB = °.
Fill in each blank
Answer:
Part B: Choose, or true or false (2 marks each)
11.OR is drawn with compasses and a straight edge, as shown (OP = OQ, PR = QR). Why does OR bisect ∠AOB?
- A. OP = OQ, PR = QR, so △OPR ≅ △OQR (RHS)
- B. ∠OPR = ∠OQR, OP = OQ, so △OPR ≅ △OQR (ASA)
- C. OP = OQ, ∠POR = ∠QOR, OR = OR, so △OPR ≅ △OQR (SAS)
- D. OP = OQ, PR = QR, OR = OR, so △OPR ≅ △OQR (SSS)
Choose the right answer
Answer:
12.To draw the perpendicular bisector of AB = 8 cm, arcs of radius 3 cm are drawn with centres A and B. They must cross.
True or false?
Answer:
Part C: Word problems (4 marks each)
13.Two fountains A and B in a park are 18 m apart. A lamp P is put on the perpendicular bisector of AB, 47 m from A. How far is P from B?
Working:
Answer:
14.A paper triangle OAB is folded so that OA lies on OB; the crease OR bisects ∠AOB. With ∠AOB = 64°: ∠AOR = °, ∠BOR = °.
Fill in each blank
Working:
Answer:
15.Where are all the points that are as far from A as from B?
- A. On the perpendicular bisector of AB
- B. On the line AB extended
- C. On the circle with centre A and radius AB
- D. At the midpoint of AB only
Choose the right answer
Working:
Answer:
16.OR bisects ∠AOB and ∠AOR = 44°, so ∠AOB = 98°.
True or false?
Working:
Answer:
17.Draw a 90° angle at a point O on a line with compasses and a straight edge: put the steps in order.
- A. PR = QR: with centres P and Q and one radius (more than 6 cm), draw arcs crossing at R
- B. OP = OQ: with centre O and radius 6 cm, draw an arc cutting the line at P and Q
- C. ∠ROP = 90°: join OR
Write the letters in the right order
Working:
Answer:
18.OR bisects ∠AOB and ∠AOR = 45°, so ∠AOB = 80°.
True or false?
Working:
Answer:
Show the answers and working
Bisecting an angle and a line segment worksheet: Answers and working
Part A: Working out and filling in
1. C → B → A
First the arc from O for P and Q, then arcs from P and Q for R, then OR; △OPR ≅ △OQR (SSS), so OR halves 98° into two of 49°.
2. B → A → C
P above AB first, then Q below, then join PQ; PQ bisects AB at right angles, AM = MB = 8 ÷ 2 = 4 cm.
3. 5 cm
The perpendicular bisector passes through AB's midpoint: AM = 10 ÷ 2 = 5 cm.
4. B → C → A
First the arc from O for P and Q, then arcs from P and Q for R, then OR; △OPR ≅ △OQR (SSS), so OR halves 70° into two of 35°.
5. 9 cm
Every point on the perpendicular bisector is as far from A as from B: PB = PA = 9 cm; AB's length isn't needed.
6. 61°
The bisector cuts the angle into two equal parts: ∠AOR = 122° ÷ 2 = 61°.
7. 7 cm
Every point on the perpendicular bisector is as far from A as from B: PB = PA = 7 cm; AB's length isn't needed.
8. 62°
The bisector cuts the angle into two equal parts: ∠AOR = 124° ÷ 2 = 62°.
9. 11, 5, 90
Points on the perpendicular bisector are as far from A as from B: PB = PA = 11 cm; it passes through the midpoint, MB = 10 ÷ 2 = 5 cm; and it is at right angles, ∠PMB = 90°.
10. 7, 5, 90
Points on the perpendicular bisector are as far from A as from B: PB = PA = 7 cm; it passes through the midpoint, MB = 10 ÷ 2 = 5 cm; and it is at right angles, ∠PMB = 90°.
Part B: Choose, or true or false
11. D. OP = OQ, PR = QR, OR = OR, so △OPR ≅ △OQR (SSS)
Three pairs of equal sides (OR common), so △OPR ≅ △OQR (SSS) and ∠POR = ∠QOR (corr. ∠s, ≅△s).
12. False
The radius must be more than half of AB, 8 ÷ 2 = 4 cm, for the arcs to cross; here it is 3 cm.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. 47 m
Every point on the perpendicular bisector is as far from A as from B: PB = PA = 47 m.
14. 32, 32
The crease halves ∠AOB into two equal angles: 64° ÷ 2 = 32°.
15. A. On the perpendicular bisector of AB
The points as far from A as from B make up the perpendicular bisector of AB.
16. False
The bisector cuts the angle in half: ∠AOB = 2 × 44°.
17. B → A → C
OR is the perpendicular bisector of PQ (OP = OQ, PR = QR), so ∠ROP = 90°.
18. False
The bisector cuts the angle in half: ∠AOB = 2 × 45°.
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