91Ó°ÊÓ

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

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

$x^2+3x−2=0$

For Exercises 23-27 write proofs in paragraph form. (Hint: You can use theorems from this section to write fairly short proofs for Exercises 23 and 24.)

Given: $\stackrel{\mathbf{→}}{\mathit{D}\mathit{P}}$bisects $\mathbf{∠}\mathit{A}\mathit{D}\mathit{E}\mathbf{;}\stackrel{\mathbf{→}}{\mathit{E}\mathit{P}}$bisects $\mathbf{∠}\mathit{D}\mathit{E}\mathit{C}$

Prove:width="26" height="24" role="math">BP→ bisects$\mathbf{∠}\mathit{A}\mathit{B}\mathit{C}$

Suppose you are given a scalene triangle and a point $P$on some line $l$. How many triangles are there with one vertex at $P$, another vertex on$l$, and each triangle congruent to the given triangle.

Draw and label a diagram. List, in terms of the diagram, what is given and what is to be proved. Then write a two-column proof.

If a line perpendicular to $\stackrel{¯}{\mathrm{AB}}$passes through the midpoint of $\stackrel{¯}{\mathrm{AB}}$, and segments are drawn from any other point on that line to and , then two congruent triangles are formed.

a. If $\mathbf{m}∠\mathbf{1}\mathbf{=}\mathbf{35}$, find$\mathbf{m}∠\mathbf{ABC}$

b. If$\mathbf{m}∠\mathbf{1}\mathbf{=}\mathbf{k}$ find $\mathbf{m}∠\mathbf{ABC}$.

Suppose you are given a scalene triangle and a point $P$on some line localid="1648799479069" $l$. How many triangles are there with one vertex at localid="1648799462577" $P$, another vertex on localid="1648799472074" $l$, and each triangle congruent to the given triangle?

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

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

${(y−5)}^2=16$

a. If$\mathbf{m}∠\mathbf{1}\mathbf{=}\mathbf{23}$ find $\mathbf{m}∠\mathbf{7}$.

b. If$\mathbf{m}∠\mathbf{1}\mathbf{=}\mathbf{k}$ find $\mathbf{m}∠\mathbf{7}$.

Draw and label a diagram. List, in terms of the diagram, what is given and what is to be proved. Then write a two-column proof.

25. If pentagon$\mathbf{ABCDE}$v is equilateral and has right angles at $\mathbf{B}$and $\mathbf{E}$, then diagonals $\stackrel{¯}{\mathrm{AC}}$and $\stackrel{¯}{\mathrm{AD}}$form congruent triangles.

For Exercises 23-27 write proofs in paragraph form. (Hint: You can use theorems from this section to write fairly short proofs for Exercises 23 and 24.)

Given: Plane $\mathit{M}$is the perpendicular bisecting plane of $\stackrel{\mathit{¯}}{\mathit{A}\mathit{B}}$, (That is, $\stackrel{\mathit{¯}}{\mathit{A}\mathit{B}}\mathbf{âŠ\yen }$plane $\mathit{M}$and $\mathit{O}$is the midpoint of$\stackrel{\mathit{¯}}{\mathit{A}\mathit{B}}$)

Prove: $$\begin{gathered} \mathrm{a}.\stackrel{\mathit{¯}}{\mathit{A}\mathit{D}}â‰\dots \stackrel{\mathit{¯}}{\mathit{B}\mathit{D}} \\ \mathrm{b}.\stackrel{\mathit{¯}}{\mathit{A}\mathit{C}}â‰\dots \stackrel{\mathit{¯}}{\mathit{B}\mathit{C}} \\ \mathrm{c}.\mathbf{∠}\mathit{C}\mathit{A}\mathit{D}\mathbf{â‰\dots }\mathbf{∠}\mathit{C}\mathit{B}\mathit{D} \end{gathered}$$