91Ó°ÊÓ

Given: $\mathrm{Plane}\mathrm{X}âˆ\yen \mathrm{Plane}\mathrm{Y};\stackrel{¯}{\mathrm{LM}}â‰\dots \stackrel{¯}{\mathrm{ON}}$

Prove: LMNO is a parallelogram.

Draw $\mathrm{Δ}\text{ABC}$ and let D be the midpoint of $\stackrel{¯}{\text{AB}}$. Let E be the midpoint of $\stackrel{¯}{\text{CD}}$. Let F be the intersection of $\stackrel{→}{\text{AE}}$ and $\stackrel{¯}{\text{BC}}$. Draw$\stackrel{¯}{\text{DG}}$ parallel to$\stackrel{¯}{\text{EF}}$ meeting$\stackrel{¯}{\text{BC}}$ at G. Prove that $\text{BG}=\text{GF}=\text{FC}$ .

Each figure in Exercises 19-24 is a parallelogram with its diagonals drawn. Find the values of x and y.

In the diagram, $\mathbf{m}∠\mathbf{VOZ}\mathbf{=}\mathbf{90}$.

$\stackrel{¯}{\mathrm{OW}}$ is an altitude of $△\mathbf{VOZ}$.

$\stackrel{¯}{\mathrm{OX}}$ bisects $∠\mathbf{VOZ}$.

$\stackrel{¯}{\mathrm{OY}}$ is a median of $△\mathbf{VOZ}$.

Find the measures of the four numbered angles.

$\mathbf{m}∠\mathbf{Z}\mathbf{=}\mathbf{30}$

Discuss how to find an alternative two-parameter family of solutions for the nonlinear differential equation y’’ = 2x( y’)2 in Example 1.

Given: Parallel planes P, Q and Rcutting transversal $\stackrel{↔}{\text{AC}}$and $\stackrel{↔}{\text{DF}}$; $\text{AB}=\text{BC}$.

Prove: $\text{DE}=\text{EF}$.

(Hint: You can’t assume that $\stackrel{↔}{\text{AC}}$and $\stackrel{↔}{\text{DF}}$are coplanar. Draw $\stackrel{¯}{\text{AF}}$, cutting plane Q at X. using the plane $\stackrel{¯}{\text{AC}}$and $\stackrel{¯}{\text{AF}}$, apply Theorems 3-1 and 5-10. Then use the plane of $\stackrel{¯}{\text{AF}}$and $\stackrel{¯}{\text{FD}}$.)

$\stackrel{¯}{\text{RA}}$is an altitude of $\mathrm{Δ}\text{SAT}$. P and Q are midpoints of $\stackrel{¯}{\text{SA}}$and $\stackrel{¯}{\text{TA}}$. $\text{SR}=9$, $\text{RT}=16$, $\text{QT}=10$, and $\text{PR}=7.5$.

Find $\text{SA}$.

Quad. DECK is a parallelogram. Complete.

If $\mathrm{KT}=2\mathrm{x}+\mathrm{y},\mathrm{DT}=\mathrm{x}+2\mathrm{y},\mathrm{TE}=12$and $\mathrm{TC}=9$, then$\mathrm{x}=?$ and$\mathrm{y}=?$

Write a paragraph proof.

Given: $â–\pm ABCD;\text{ â¶Ä‰}â–\pm BEDF$

Prove: AECF is a $â–\pm$.

(Hint: A short proof is possible if certain auxiliary segments ate drawn.)

In the following figure is rhombus $\stackrel{¯}{RA}$is an altitude $\mathrm{Δ}SAT$. Pand Q are midpoint of $\stackrel{¯}{SA}$and $\stackrel{¯}{TA}$. $SR=9,RT=16,QT=10,PR=7.5$. Find $SA$.