91Ó°ÊÓ

In Exercises 13 and \(14,\) write a proof to verify that the construction is valid. Line perpendicular to a line through a point not on the line Plan for Proof Show that \(\triangle \mathrm{APQ} \cong \triangle \mathrm{BPQ}\) by the SSS Congruence Theorem (Theorem 5.8 ). Then show that \(\triangle \mathrm{APM} \cong \triangle \mathrm{BPM}\) using the SAS Congruence Theorem (Theorem 5.5). Use corresponding parts of congruent triangles to show that \(\angle \mathrm{AMP}\) and \(\angle \mathrm{BMP}\) are right angles.

In Exercises 13 and \(14,\) graph the triangle with the given vertices. Find the length and the slope of each side of the triangle. Then nd the coordinates of the midpoint of each side. Is the triangle a right triangle? isosceles? Explain. (Assume all variables are positive and \(\quad \mathrm{m} \neq \mathrm{n} . )\) (See Example 3.) $$ \mathrm{D}(0, \mathrm{n}), \mathrm{E}(\mathrm{m}, \mathrm{n}), \mathrm{F}(\mathrm{m}, 0) $$

ERROR ANALYSIS In Exercises 15 and 16, describe and correct the error. \(\Delta Q R S \cong \Delta V W X\) by the \(A A S\) Congruence Theorem.

In Exercises 15 and 16, write a proof. Given \(\quad \overline{\mathrm{WX}} \cong \overline{\mathrm{VZ}}, \overline{\mathrm{WY}} \cong \overline{\mathrm{VY}}, \overline{\mathrm{YZ}} \cong \overline{\mathrm{YX}}\) Prove \(\quad \Delta \mathrm{VWX} \cong \Delta \mathrm{WVZ}\)

PROVING A THEOREM Prove the Third Angles Theorem (Theorem 5.4) by using the Triangle Sum Theorem (Theorem 5.1).

THOUGHT PROVOKING Draw a triangle. Copy the triangle multiple times to create a rug design made of congruent triangles. Which property guarantees that all the triangles are congruent?

THOUGHT PROVOKINGhe Bermuda Triangle is a region in the Atlantic Ocean in which many ships and planes have mysteriously disappeared. The vertices are Miami, San Juan, and Bermuda. Use the Internet or some other resource to I nd the side lengths, the perimeter, and the area of this triangle (in miles). Then create a congruent triangle on land using cities as vertices.

Find the perimeter of the polygon with the given vertices. $$ A(-1,1), B(4,1), C(4,-2), D(-1,-2) $$

In Exercises \(23-26,\) ind the measure of each acute angle in the right triangle. The measure of one acute angle is 8 times the measure of the other acute angle.

In Exercises \(23-26,\) ind the measure of each acute angle in the right triangle. The measure of one acute angle is twice the difference of the measure of the other acute angle and 12 .