Question

In Circle P, m EA=58, m BC=42, m CD=90, find:

Answer 1

Circle theorem states that the angle that an arc subtends at the center of a

circle is twice the angle subtended at the circumference.

The values are as follows;

1. m∠EDA = 90°
2. m∠BDC = 21°
3. m∠BEC = 21°
4. $m\widehat{AB}$ = 80°
5. m∠AEB = 40°
6. m∠ECA =  29°
7. m∠BAC = 21°
8. $m\widehat{ED}$  =  90°
9. m∠ECD = 45°
10. m∠ACB = 40°
11. m∠EBD = 45°
12. m∠ADB = 40°
13. m∠CAD = 45°
14. m∠EAB = 90°
15. m∠EBC = 90°

Reasons:

$\displaystyle 1. \ \mathbf{m\angle EDA} = \frac{1}{2} \cdot m\angle EPC$ Angle subtended at center is twice angle at circumference.

m∠EPC = 180°

∴  m∠EDA = 0.5 × 180° = 90°

2. $m\widehat{BC}$ = 42°, therefore, m∠BPC = 42°

$\displaystyle \ m\angle BDC= \frac{1}{2} \cdot m\angle BPC$

m∠BDC = 0.5 × 42° = 21°

3. m∠BEC = m∠BDC = 21°

4. $m\widehat{AB}$ = 180° - (58° + 42°) = 80°

5. $\mathbf{m\widehat{AB}}$ = m∠APB by definition

m∠AEB = 0.5 × m∠APB

m∠AEB = 0.5 × 80° = 40°

6. m∠ECA = 0.5 × m∠AE

Therefore;

m∠ECA = 0.5 × $m\widehat{EA}$

m∠ECA = 0.5 × 58° = 29°

m∠ECA =  29°

7. m∠BAC = 0.5 × $\mathbf{m\widehat{BC}}$

m∠BAC = 0.5 × 42° = 21°

m∠BAC = 21°

8. $m\widehat{ED}$  = 180° - $\mathbf{m\widehat{CD}}$

$m\widehat{ED}$  = 180° - 90° = 90°

$\mathbf{m\widehat{ED}}$  =  90°

9. m∠ECD = 0.5 × $\mathbf{m\widehat{ED}}$

m∠ECD = 0.5 × 90° = 45°

10. m∠ACB = 0.5 × $m\widehat{AB}$

m∠ACB = 0.5 × 80° = 40°

11. m∠EBD = 0.5 × $\mathbf{m\widehat{ED}}$

m∠EBD = 0.5 × 90° = 45°

12. m∠ADB = 0.5 × $\mathbf{m\widehat{AB}}$

m∠ADB = 0.5 × 80° = 40°

13. m∠CAD = 0.5 × $\mathbf{m\widehat{CD}}$

m∠CAD = 0.5 × 90° = 45°

14. m∠EAB = 0.5 ×$\mathbf{m\widehat{EABC}}$

m∠EAB = 0.5 × 180° = 90°

15. m∠EBC = 0.5 ×$\mathbf{m\widehat{EABC}}$

m∠EBC = 0.5 × 180° = 90°

Learn more about circle theorems here:

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