Step-5

Title: Congruency of triangles(SAS)

Grade: 10-a Lesson: S2-L5

Explanation:

Description: 5

d5.2

Steps Statment Solution

1

Given

ABCD is a square, the diagonals of the square intersects at E and \$\angle ABE = 45^\circ\$

2

Consider \$\triangle AEB\$ and \$\triangle CED\$

\begin{align} AE &= CE \\ \angle AEB &= \angle CED \\ BE &= DE \\ \therefore \triangle AEB & \cong \triangle CED \text{(by SAS congruency)} \end{align}

3

Consider \$\triangle AEB\$

\begin{align} \angle ABE + \angle BEA + \angle BAE &= 180^\circ \\ 45^\circ + 90^\circ + \angle BAE &= 180^\circ \\ 135^\circ + \angle BAE &= 180^\circ \\ \angle BAE &= 180^\circ - 135^\circ \\ \angle BAE &= 45^\circ \\ \end{align}

4

From \$\triangle AEB\$ and \$\triangle CED\$

\begin{align} \triangle AEB &\cong \triangle CED \\ \angle EAB &= \angle ECD \\ 45^\circ &= \angle ECD \\ \end{align}

Given,

d5.2

ABCD is a square, the diagonals of the square intersects at E and \$\angle ABE = 45^\circ\$.

In a square the diagonals bisect each other perpendicularly and equally, also the angles formed by the diagonals at the point of intersection are \$90^\circ\$.

Now in square ABCD, AC and BD are the diagonals intersecting at point E.

Angles formed by the diagonals are \$\angle AEB = \angle AED = \angle DEC = \angle CEB = 90^\circ\$.

Consider \$\triangle AEB\$

\begin{align} \angle ABE + \angle BEA + \angle BAE &= 180^\circ \\ 45^\circ + 90^\circ + \angle BAE &= 180^\circ \\ 135^\circ + \angle BAE &= 180^\circ \\ \angle BAE &= 180^\circ - 135^\circ \\ \angle BAE &= 45^\circ \\ \end{align}

Consider \$\triangle AEB\$ and \$\triangle CED\$

\begin{align} AE &= CE \\ \angle AEB &= \angle CED \\ BE &= DE \\ \therefore \triangle AEB & \cong \triangle CED \text{(by SAS congruency)} \end{align}

From \$\triangle AEB\$ and \$\triangle CED\$

\begin{align} \triangle AEB &\cong \triangle CED \\ \angle EAB &= \angle ECD \\ 45^\circ &= \angle ECD \\ \end{align}

\begin{align} \therefore \angle ECD = 45^\circ \\ \end{align}


Copyright © 2020-2022 saibook.us Contact: info@saibook.us Version: 1.5 Built: 01-December-2022 05:00 PM EST