Chapter 4: Congruent Triangles

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 ?.

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

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.

For the following figure, (a) List two pairs of congruent corresponding sides and one pair of congruent corresponding angles in $\Delta YTR$and $\Delta XTR$. (b) Notice that, in each triangle, you listed two sides and nonincluded angle. Do you think that SSA is enough to guarantee that two triangles are congruent?

Plane M is the perpendicular bisecting plane of at O (that is, M is the plane that is perpendicular to 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.

Write proofs in two–column form.

Theorem 4–1.

Decide whether you can deduce by the SSS, SAS, or ASA postulate that another triangle is congruent to $\Delta ABC$. If so, write the congruence and name the postulate used. If not, write no congruence can be deduced.

Complete each statement.

It $O$is on the perpendicular bisector of $\overline{AF}$, then $O$is equidistant from$\underline{?}$ and $\underline{?}$.

Given:$\overline{WX}\perp \overline{UV};\overline{WX}\perp \overline{YZ};\overline{WU}\cong \overline{WV}$

Prove whatever you can about angles 1,2,3, and 4

Explain how corollary follows from theorem 4-2.