Question

triangle congruence theorems
prove whether the given triangles are congruent. use the triangle congruence theorems to justify your answer.

1. prove △stv≅△ead.
2. prove △lpm≅△qts.
3. prove △abc≅△dcb.
4. determine if each equation is a correct step in the process of solving -9x + 11=-x² - 7 for x. select yes or no for each equation.

yes no
x² - 9x + 4 = 0
x² - 9x + 18 = 0
(x + 6)(x + 3)=0
(x - 6)(x - 3)=0

Explanation:

1. Prove $\triangle STV\cong\triangle EAD$

Given $\angle STV=\angle EAD$, $\angle TVS = \angle ADE$, $SV = EA$. By the Angle - Angle - Side (AAS) congruence theorem, since two angles and a non - included side of one triangle are equal to the corresponding two angles and non - included side of the other triangle, $\triangle STV\cong\triangle EAD$.

2. Prove $\triangle LPM\cong\triangle QTS$

Given $\angle LPM=\angle QTS$, $LP = QT$, $\angle PML=\angle TSQ$. By the Angle - Side - Angle (ASA) congruence theorem, as two angles and the included side of one triangle are equal to the corresponding two angles and included side of the other triangle, $\triangle LPM\cong\triangle QTS$.

3. Prove $\triangle ABC\cong\triangle DCB$

We have $AB = DC$, $BC=CB$ (common side), $AC = DB$. By the Side - Side - Side (SSS) congruence theorem, since all three sides of one triangle are equal to the corresponding three sides of the other triangle, $\triangle ABC\cong\triangle DCB$.

4. Solve the equation $-9x + 11=-x^{2}-7$

Step1: Rearrange the equation

Starting with $-9x + 11=-x^{2}-7$, add $x^{2}$ and 7 to both sides of the equation.
$x^{2}-9x + 11+7=0$, which simplifies to $x^{2}-9x + 18 = 0$. So the statement $x^{2}-9x + 4=0$ is No.

Step2: Factor the quadratic equation

For the quadratic equation $x^{2}-9x + 18 = 0$, we need to find two numbers that multiply to 18 and add up to - 9. The numbers are - 6 and - 3.
We can factor it as $(x - 6)(x - 3)=0$. So the statements $(x + 6)(x + 3)=0$ is No and $(x - 6)(x - 3)=0$ is Yes.

Answer:

1. $\triangle STV\cong\triangle EAD$ by AAS.
2. $\triangle LPM\cong\triangle QTS$ by ASA.
3. $\triangle ABC\cong\triangle DCB$ by SSS.

4.

Equation	Answer
$x^{2}-9x + 18=0$	Yes
$(x + 6)(x + 3)=0$	No
$(x - 6)(x - 3)=0$	Yes