Chapter 4: Congruent Triangles

Plane M is the perpendicular bisecting plane of $\overline{\mathit{A}\mathit{B}}$ at O (that is, M is the plane that is perpendicular to $\overline{\mathit{A}\mathit{B}}$ at its midpoint, O). Points C and D also lie in plane M. List three pairs of congruent triangles and tell which congruence method can be used to prove each pair congruent.

Suppose you know that $\overline{DB}\perp \overline{DC}.$Explain how you can deduce that $\overline{DB}\perp \overline{BA}.$

Tell which pairs of congruent parts and what methods $\mathbf{(}\mathit{S}\mathit{S}\mathit{S}\mathbf{,}\mathbf{}\mathit{S}\mathit{A}\mathit{S}\mathbf{,}\mathbf{}\mathit{A}\mathit{S}\mathit{A}\mathbf{,}\mathbf{}\mathit{A}\mathit{A}\mathit{S}\mathbf{,}\mathbf{}\mathbf{or}\mathbf{}\mathit{H}\mathit{L}\mathbf{)}$you would use to prove the triangles are congruent.

Given: $\angle \mathbf{1}\cong \angle \mathbf{2}\mathbf{;}\mathbf{}\mathbf{\angle }\mathbf{3}\mathbf{\cong }\mathbf{\angle }\mathbf{4}\mathbf{;}\mathbf{}\overline{\mathrm{QR}}\mathbf{\cong }\overline{\mathrm{TS}}$

$\mathbf{△}\mathbf{QPR}\mathbf{\cong }\mathbf{△}\mathbf{TPS}$by what method?

Find the measure of each numbered angle.

Given:$\overline{WX}\perp \overline{YZ};\angle 1\cong \angle 2;\overline{UX}\cong \overline{VX}$

Which one(s) of the following statements must be true?

(1)$\overline{XW}\perp \overline{UV}$

(2)$\overline{UV}\parallel \overline{YZ}$

(3)$\overline{VX}\perp \overline{UX}$

Write proofs in the form specified by your teacher (two-column form, paragraph form, or a list of key steps).

Given: $\overline{\mathrm{DE}}\cong \overline{\mathrm{FG}};\overline{\mathrm{GD}}\cong \overline{\mathrm{EF}}$;

$\angle \mathbf{HDE}\mathbf{and}\angle \mathbf{KFG}\mathbf{are}\mathbf{}\mathbf{rt}\mathbf{.}\angle \mathbf{s}$

Prove: $\overline{\mathrm{DH}}\cong \overline{\mathrm{FK}}$

Explain how corollary 3 follows from theorem 4-1.

The two triangles shown are congruent. Complete.

a. $\mathrm{\Delta }PAL\cong$? .

b. $\overline{PA}\cong$?.

c. $\angle \mathbf{1}\cong$? because ? .

Then $\overline{PA}\parallel \underset{¯}{?}$ because ?.

d. $\angle \mathbf{2}\cong \underset{¯}{?}$ because ?.

Then $\underset{¯}{?}\parallel \underset{¯}{?}$ because ?.

Solve each equation by factoring or by using the quadratic formula. The quadratic formula is:

If $a{x}^2+bx+c=0$, with $a\ne 0$, then $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$.

$y^2-6y+9=0$

Given that $\overline{\mathbf{B}\mathbf{C}}\mathbf{\cong }\overline{\mathbf{C}\mathbf{D}}\mathbf{,}\mathbf{}\overline{\mathbf{B}\mathbf{D}}\mathbf{\cong }\overline{\mathbf{C}\mathbf{E}}$then prove that$\mathbf{\Delta }\mathit{A}\mathit{B}\mathit{C}$is isosceles.