Question

9.
calculate ( mangle def ).
a) ( 180^circ ) b) ( 80^circ )
c) ( 120^circ ) d) ( 60^circ )

1. ( overrightarrow{bd} ) bisects ( angle abc ). find the value of ( x ).

a) ( x = 55 ) b) ( x = 43 )
c) ( x = 12 ) d) ( x = 67 )

Explanation:

Question 9

Step1: Identify ∠DEF

∠DEF is the angle at E between D and F. From the diagram, the angle labeled for ∠DEF (between E to F and the straight line) is 120°, or we can check the straight line: D - E - F is a straight line (180°), but the angle given for ∠DEF (the larger angle) is 120° as per the diagram.

Step2: Match with options

The options are a) 180°, b) 80°, c) 120°, d) 60°. So the measure of ∠DEF is 120°.

Step1: Use Angle Bisector Property

Since \(\overrightarrow{BD}\) bisects \(\angle ABC\), \(\angle ABD=\angle CBD\). Given \(\angle CBD = 55^\circ\), so \(\angle ABD=(x + 12)^\circ\) should equal \(55^\circ\)? Wait, no: Wait, the total angle? Wait, no, bisector means it splits ∠ABC into two equal angles. Wait, ∠ABC is split into ∠ABD and ∠CBD. So \(\angle ABD=\angle CBD\). Wait, the diagram shows ∠CBD is 55°, and ∠ABD is \((x + 12)^\circ\). Wait, no, maybe ∠ABC is composed of ∠ABD and ∠CBD, and BD bisects it, so ∠ABD = ∠CBD? Wait, no, maybe ∠ABC is 2∠ABD (since BD bisects it). Wait, no, let's re - read: \(\overrightarrow{BD}\) bisects \(\angle ABC\), so \(\angle ABD=\angle CBD\). Wait, the diagram: at point B, we have BA, BD, BC. So ∠ABC is the angle between BA and BC, and BD bisects it, so ∠ABD = ∠CBD. Given ∠CBD = 55°, so ∠ABD = 55°? Wait, no, the label for ∠ABD is \((x + 12)^\circ\), and ∠CBD is 55°. Wait, maybe I got it wrong. Wait, no, maybe ∠ABC is ∠ABD + ∠CBD, and BD bisects it, so ∠ABD = ∠CBD. Wait, no, if BD bisects ∠ABC, then ∠ABD = ∠DBC. So \((x + 12)^\circ=55^\circ\)? No, that would give x = 43, but wait, maybe ∠ABC is 2∠ABD, and ∠ABC is ∠ABD + ∠DBC, with ∠DBC = 55°, so ∠ABD = 55°, so \(x + 12=55\)? No, 55 - 12 = 43? Wait, no, maybe the total angle? Wait, no, let's do the math: If BD bisects ∠ABC, then ∠ABD = ∠CBD. So \((x + 12)^\circ=55^\circ\)? No, that would be x = 43, but let's check the options. Option b is x = 43. Wait, let's solve \(x + 12=55\)? No, 55 - 12 = 43. Wait, maybe ∠ABC is 2*∠ABD, and ∠ABC is ∠ABD + ∠DBC, with ∠DBC = 55°, so ∠ABD = 55°, so \(x + 12 = 55\), so x = 55 - 12 = 43.

Step2: Solve for x

We have the equation \(x+12 = 55\) (since BD bisects ∠ABC, so ∠ABD = ∠CBD = 55°). Subtract 12 from both sides: \(x=55 - 12=43\).

Answer:

c) 120°

Question 10