CPM Homework Help : CCG Problem 7-69

Jester started to prove that the triangles at right are congruent. He was only told that point $E$ is the midpoint of segments $\overline{AC}$ and $\overline{BD}$ .

Copy and complete his flowchart below. Be sure that a reason is provided for every statement.

Two lines A, C and B, D intersect at point, E. Triangle A, B, ,E and D, C, E are formed.

	$\enclose{circle}{\; \\ \qquad \text{E is a midpoint} \qquad \\}$
	$\huge \swarrow$ Bubble 1 has an arrow pointing down to bubble 3.	$\huge \searrow$ Given Bubble 1 has an arrow pointing down to bubble 4.
$\enclose{circle}{\; \\ \qquad \angle AEB \cong \angle CED \qquad \\}$ Angle A, E, B, is congruent to angle C, E, DVertical angles are congruent.	$\enclose{circle}{\; \\ \qquad \ \ \ \qquad \qquad \\}$ Fill in bubble 3.	$\enclose{circle}{\; \\ \qquad \overline{AE} \cong \overline{CE} \qquad \\}$ A, E is congruent to C, E.Definition of midpoint.
$\huge \searrow$ Bubble 2 has an arrow pointing down to bubble 5.	$\huge \downarrow$ Bubble 3 has an arrow pointing down to bubble 5.	$\huge \swarrow$ Bubble 4 has an arrow pointing down to bubble 5.
	$\enclose{circle}{\; \\ \qquad \ \ \ \qquad \qquad \\}$ Fill in bubble 5.

Answer:

	$\enclose{circle}{\; \\ \qquad \text{E is a midpoint} \qquad \\}$
	$\huge \swarrow$ Bubble 1 has an arrow pointing down to bubble 3.	$\huge \searrow$ Given Bubble 1 has an arrow pointing down to bubble 4.
$\enclose{circle}{\; \\ \qquad \angle AEB \cong \angle CED \qquad \\}$ Vertical angles are congruent.	$\enclose{circle}{\; \\ \qquad \bf{\color{red}{\overline{BE}\cong \overline{DE}}} \qquad \\}$ $\bf{\color{red}{\text{Definition of midpoint.}}}$	$\enclose{circle}{\; \\ \qquad \overline{AE} \cong \overline{CE} \qquad \\}$ Definition of midpoint.
$\huge \searrow$ Bubble 2 has an arrow pointing down to bubble 5.	$\huge \downarrow$ Bubble 3 has an arrow pointing down to bubble 5.	$\huge \swarrow$ Bubble 4 has an arrow pointing down to bubble 5.
	$\enclose{circle}{\; \\ \qquad \bf{\color{red}{\triangle AEB \cong \triangle CED}} \qquad \\}$ $\quad \quad \quad \quad \quad \quad \quad \bf{\color{red}{SAS \cong}}$