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Chapter 3: Q31 (page 99)

Given: AB⊥BF;HD⊥BFand GF⊥BF.

∠A≅∠G

Which numbered angles must be congruent?

Short Answer

Expert verified

The congruent angles are ∠1≅∠2,∠3≅∠4and ∠4≅∠6.

Step by step solution

01

Step 1. Apply definition of vertically opposite angles.

Vertically opposite angles are the opposite angles formed due to intersection of two lines.

Consider ΔHDCandΔABC,

∠1and∠2are vertically opposite angles, therefore, ∠1≅∠2.

02

Step 2. Description of step.

HD¯⊥BF¯and FG¯⊥BF¯implies that HD¯||FG¯, that is, if two lines are perpendicular to same line then they are parallel to each other.

When two lines are intersected by a transversal then the angles formed on the opposite sides of a transversal with respect to two lines are congruent. Therefore,

∠4≅∠6

03

Step 3. Apply the definition of angle bisector.

Angle bisector is a line which divides an angle into two equal parts.

From the figure, it can be observed that HD¯bisect ∠CHE. Therefore,

∠3≅∠4

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∠3≅∠6
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