Lesson Example Discussion Quiz: Class Homework |
Step-5 |
Title: Congruency of triangles(SAS) |
Grade: 10-a Lesson: S2-L5 |
Explanation: |
Description: 5
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,
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}
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