Math / Geometry

Question(b) In $\triangle \mathrm{PQR}, \mathrm{TP} \| \mathrm{QRS}$ . (1)* Express $y$ in terms of $\theta$ . (2) Calculate the size of $\theta$ . (c) In quadrilateral $\mathrm{ABCE}, \hat{\mathrm{C}}_{1}=75^{\circ}$ , $\hat{\mathrm{B}}_{1}=20^{\circ}, \hat{\mathrm{E}}_{2}=x, \hat{\mathrm{F}}_{1}=y$ , $\mathrm{AF}=\mathrm{BF}$ and $\mathrm{FE}=\mathrm{FC}$ . (1) Calculate the size of $x$ and $y$ . (2) Prove that $A B \| E C$ . (d) ABCD is a parallelogram with $\mathrm{AB}=\mathrm{AE}=\mathrm{EC}$ and $\hat{\mathrm{E}}_{1}=60^{\circ}$ . Calculate the size of: (1) $x$ (2) $\hat{\mathrm{E}}_{2}$

Studdy Solution
To find $\hat{\mathrm{E}}_{2}$ : Since $\mathrm{ABCD}$ is a parallelogram, $\hat{\mathrm{E}}_{2}$ is the exterior angle to $\angle DAB$ , $\hat{\mathrm{E}}_{2} = \angle DAB = 180^\circ - \angle E_1$ $\hat{\mathrm{E}}_{2} = 180^\circ - 60^\circ$ $\hat{\mathrm{E}}_{2} = 120^\circ$
Final Solution:
1. $y = 60^\circ$
2. $\theta = 50^\circ$
3. $x = 52.5^\circ$ , $y = 80^\circ$
4. $\mathrm{AB} \parallel \mathrm{EC}$ is not possible.
5. $x = 60^\circ$
6. $\hat{\mathrm{E}}_{2} = 120^\circ$

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