Question

1. in the diagram, $\triangle abc \cong \triangle dec$. find the values of $x$ and $y$.

1. in the diagram, $rstu \cong uvqr$. find the values of $x$ and $y$.

Explanation:

Step1: Identify congruent parts for Q2

Corresponding angles: $\angle A = \angle D = 34^\circ$, $\angle ACB = \angle DCE = 81^\circ$ (vertical angles). Corresponding sides: $AB = DE = 13$, $BC = EC = 6$.

Step2: Calculate x for Q2

Sum of angles in $\triangle ABC$: $x = 180^\circ - 34^\circ - 81^\circ = 65^\circ$.

Step3: Calculate y for Q2

Sum of angles in $\triangle DEC$: $y = 180^\circ - 34^\circ - 81^\circ = 65^\circ$.

Step4: Analyze congruent quadrilaterals for Q3

$RSTU \cong UVQR$: Corresponding angles $\angle URS = \angle QUV = 47^\circ$, so $(2x + 5y)^\circ = 47^\circ$. Right angles imply $(4x - 3y)^\circ = 90^\circ$.

Step5: Solve equations for Q3

Equation1: $2x + 5y = 47$; Equation2: $4x - 3y = 90$. Multiply Equation1 by 2: $4x + 10y = 94$. Subtract Equation2: $13y = 4 \Rightarrow y = 4$. Substitute $y=4$ into Equation1: $2x + 20 = 47 \Rightarrow 2x=27 \Rightarrow x=13.5$.

Answer:

Q2: $x=65$, $y=65$
Q3: $x=13.5$, $y=4$