Exam-Style Problems

Stage 9 - Q1 - 2 marks

19427

The internal bisectors from B and C meet at the marked point \(I\). Find angle \(BIC\).

\[\angle A=56^\circ\]

A

B

C

Hint

Use the angle sum in the smaller triangle \(BIC\).

Stage 9 - Q2 - 2 marks

19428

B and C are bisected; I is their marked intersection. Find angle \(BIC\) from the two given original angles.

\[\angle B=44^\circ,\quad\angle C=76^\circ\]

A

B

C

Hint

The angles in the smaller triangle are halves of B and C.

Stage 9 - Q3 - 3 marks

19429

Internal bisectors from B and C meet at I. Find the original angle \(A\).

\[\angle BIC=122^\circ\]

A

B

C

Hint

Subtract the right angle, then double.

Stage 9 - Q4 - 3 marks

19430

Two triangles have the same angle \(A=68^\circ\) but their B angles are \(54^\circ\) and \(55^\circ\). In each, bisectors of B and C meet at I. Must the angles \(BIC\) match? Explain.

A

B

C

Hint

The sum \(B+C\) stays fixed even when its parts change.

Stage 9 - Q5 - 4 marks

19431

Extend the two bisectors through marked I. Find the acute and obtuse angles where the lines cross.

\[\angle A=72^\circ\]

A

B

C

Hint

Do not confuse the obtuse internal angle with its adjacent acute angle.

Stage 9 - Q6 - 4 marks

19432

Extend the two bisectors through marked I. Find the acute and obtuse angles where the lines cross.

\[\angle A=76^\circ\]

A

B

C

Hint

Do not confuse the obtuse internal angle with its adjacent acute angle.

Stage 9 - Q7 - 5 marks

19433

A learner predicts the bisector angle \(BIC\) as \(180-\frac{A}{2}\) when \(A=80^\circ\). Explain the error and correct the result.

A

B

C

Hint

Write the angle sum for triangle \(BIC\) before simplifying.

Stage 9 - Q8 - 5 marks

19434

All three internal bisectors meet at I. Find the three original triangle angles from the two given intersection angles, then find angle AIB.

\[\angle BIC=132^\circ\]

\[\angle CIA=118^\circ\]

Hint

Match each bisector angle with the vertex opposite its two named endpoints.

Stage 9 - Q9 - 5 marks

19435

The internal bisectors from B and C meet at I. The angle \(BIC\) is 7 times angle \(A\). Find angle \(A\) and angle \(BIC\) exactly.

Hint

The constant 90 remains when the two expressions are equated.

Stage 9 - Q10 - 3 marks

19436

Internal bisectors from B and C meet at the marked point \(I\). Angle B is twice angle \(A\), and angle \(BIC\) is three times angle \(A\). Find all triangle angles, angle \(BIC\) and the adjacent acute crossing angle.

A

B

C

Hint

The condition on angle \(BIC\) determines A before the condition on B is needed.