CPM Homework Help : GC Problem 11-39

Quadrilateral, A, B, C, D, line segments from A, to C, from, B, to, D, intersecting at, e. In the diagram at right, assume that $m∠ECB = m∠EAD$ and point $E$ is the midpoint of $\overline { A C }$ . Prove that $\overline { A D }≅\overline{CB}$ .

Hint:

If you can prove that $\Delta DEA \cong \Delta CEB$ then you can prove $\overline{AD} \cong \overline{CB}$ .

More Help:

You can prove that $ΔDEA ≅ ΔCEB$ by $\text{AAS} ≅$ .

Answer:

Flowchart, 6 bubbles, Bubbles 1, 2, 3 on top row, 4, & 5 on middle row, and 6, on bottom row, with arrows as follows: from 1 to 4. From 2 to 4, from 3 to 5, from 5 to 4, from 4 to 6.

Bubble 1 $\enclose{circle}{ \; \\ \quad m \angle ECB = m \angle EAD \quad \\}$	Bubble 2 $\enclose{circle}{ \; \\ \quad m \angle AED = m \angle CEB \quad \\}$		Bubble 3 $\enclose{circle}{ \; \\ \; \\ \quad E \; \text{is the midpoint} \quad \\ \qquad \quad \text{of} \; \overline{AC}\\}$
$\text{Given} \; \Huge \searrow$	${\text{Vertical} \; \\ \text{angles are} \; \\ \text{congruent}}\; \Huge \downarrow$		${\Huge \downarrow} \; \text{Given}$
$\text{Given} \; \Huge \searrow$	Bubble 4 $\enclose{circle}{ \; \\ \quad \Delta AED \; \cong \; \Delta CEB \quad \\}$	$\Huge \leftarrow$	Bubble 5 $\begin{array}{c} \enclose{circle}{ \; \\ \quad \overline{AE} \; \cong \; \overline{CE} \quad \\} \\ \text{Def. of midpoint} \end{array}$
	${\Huge \downarrow} \; \text{ASA}\cong$
	Bubble 6 $\begin{array}{l} \enclose{circle}{ \; \\ \quad \overline{AD} \; \cong \; \overline{CB} \quad \\} \\ \qquad \qquad \quad \cong \Delta \text{s} \; \rightarrow \; \; \cong \text{parts} \end{array}$