Question

1. proof complete the proof.

given: b is the mid - point of $overline{ae}$. b is the mid - point of $overline{cd}$.
prove: $\triangle abdcong\triangle ebc$
statements reasons

1. b is the mid - point of $overline{ae}$. 1.
2. 2. definition of midpoint
3. b is the mid - point of $overline{cd}$. 3.
4. 4. definition of midpoint
5. $angle abdcongangle ebc$ 5.
6. $\triangle abdcong\triangle ebc$ 6.
7. proof complete the proof.

given: $overline{ab}paralleloverline{cd}$, $overline{ab}congoverline{cd}$
prove: $\triangle abccong\triangle dcb$
statements reasons

1. $overline{ab}paralleloverline{cd}$ 1.
2. $angle abccongangle dcb$ 2.
3. $overline{ab}congoverline{cd}$ 3.
4. $overline{cb}congoverline{cb}$ 4.
5. $\triangle abccong\triangle dcb$ 5.
6. ★ multiple choice which triangle congruence can you prove, then use to prove that $angle fedcongangle abf$?

(a) $\triangle abecong\triangle abf$ (c) $\triangle aedcong\triangle abd$
(b) $\triangle acdcong\triangle adf$ (d) $\triangle aeccong\triangle abd$

1. challenge describe how to show that $\triangle pmocong\triangle pmn$ using the sss congruence postulate. then show that the triangles are congruent using the sas congruence postulate without measuring any angles. compare the two methods.

Explanation:

Step1: Problem 12

Statements

1. $B$ is the mid - point of $\overline{AE}$. Reason: Given.
2. $AB = EB$. Reason: Definition of mid - point.
3. $B$ is the mid - point of $\overline{CD}$. Reason: Given.
4. $DB=CB$. Reason: Definition of mid - point.
5. $\angle ABD=\angle EBC$. Reason: Vertical angles are congruent.
6. $\triangle ABD\cong\triangle EBC$. Reason: Side - Angle - Side (SAS) congruence postulate.

Step2: Problem 13

Statements

1. $\overline{AB}\parallel\overline{CD}$. Reason: Given.
2. $\angle ABC=\angle DCB$. Reason: Alternate interior angles are congruent for parallel lines $\overline{AB}$ and $\overline{CD}$ with transversal $\overline{BC}$.
3. $\overline{AB}\cong\overline{CD}$. Reason: Given.
4. $\overline{CB}\cong\overline{CB}$. Reason: Reflexive property of congruence.
5. $\triangle ABC\cong\triangle DCB$. Reason: Side - Angle - Side (SAS) congruence postulate.

Step3: Problem 14

In the given figure, we can prove $\triangle AED\cong\triangle ABD$.
We know that $\angle ADE = 70^{\circ}$, $\angle ADB=40^{\circ}$, and we have side - side - side or side - angle - side relationships. If we consider the common side $\overline{AD}$, and the other given side lengths and angle measures, we can show congruence. Once $\triangle AED\cong\triangle ABD$, then $\angle FED=\angle ABF$ because corresponding parts of congruent triangles are congruent. So the answer is C.

Step4: Problem 15

SSS Congruence Postulate

1. Using the distance formula $d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}$ in the coordinate - plane:

• For $\overline{PM}$, if $P(x_1,y_1)$ and $M(x_2,y_2)$, calculate the length.
• For $\overline{MO}$, if $M(x_2,y_2)$ and $O(x_3,y_3)$, calculate the length.
• For $\overline{PO}$, if $P(x_1,y_1)$ and $O(x_3,y_3)$, calculate the length.
• Do the same for the sides of $\triangle PMN$: $\overline{PM}$, $\overline{MN}$, $\overline{PN}$. Show that $PM = PM$ (reflexive), $MO=MN$, and $PO = PN$. Then by the SSS congruence postulate, $\triangle PMO\cong\triangle PMN$.

SAS Congruence Postulate

1. First, find the lengths of $\overline{PM}$ (common side for both triangles).
2. Then find the lengths of $\overline{MO}$ and $\overline{MN}$, and show that $MO = MN$.
3. Since $\overline{PM}$ is common, and we have shown the equality of the other pair of sides, and the included angle $\angle OPM=\angle NPM$ (by construction or because of the symmetry in the coordinate - plane), by the SAS congruence postulate, $\triangle PMO\cong\triangle PMN$.
4. Comparison: The SSS postulate requires finding the lengths of all three sides of each triangle. The SAS postulate requires finding the lengths of two sides and the measure of the included angle. SAS may be quicker if the angle measure is easily determined or given, while SSS is more straightforward when only side - length information is available or easily calculated.

Answer:

1. Statements: 2. $AB = EB$; 4. $DB = CB$; 5. Vertical angles are congruent; 6. SAS.
2. Reasons: 1. Given; 2. Alternate interior angles for parallel lines; 3. Given; 4. Reflexive property; 5. SAS.
3. C. $\triangle AED\cong\triangle ABD$
4. See above for SSS and SAS methods and comparison.