91Ó°ÊÓ

Accurately draw each triangle described. Predict whether the triangle will be congruent with your classmates.

a. In $\mathit{\text{â–³R°Õ³§,}}$$\mathit{\text{RS}}=4\text{cm}$, $\mathit{\text{³¾âˆ S=45}}$, and $\mathit{\text{ST}}=6\text{cm}$.

b. In $△\mathit{\text{UVW}}$, $\mathit{\text{m}}\mathit{∠}\mathit{\text{U}}=30$, $\mathit{\text{UV}}=5\text{cm}$, and $\mathit{\text{m}}\mathit{∠}\mathit{\text{V}}=100$.

c. In $\mathit{\text{â–³D·¡¹ó}}$, $\mathit{\text{m}}\mathit{∠}\mathit{\text{D}}=30$, $\mathit{\text{m}}\mathit{∠}\mathit{\text{E}}=68$, and $\mathit{\text{m}}\mathit{∠}\mathit{\text{F}}=82$.

d. In $\mathrm{Δ}\mathit{\text{XYZ}}$, role="math" localid="1648817435970" $\mathit{\text{XY}}=3\text{cm}$, $\mathit{\text{YZ}}=5\text{cm}$, and $\mathit{\text{XZ}}=6\text{cm}$. (Try for a reasonably accurate drawing. You may find it helpful to cut a thin strip of paper for each side, then form a triangle.)

Given: $\stackrel{¯}{\mathrm{PO}}â‰\dots \stackrel{¯}{\mathrm{QO}}\mathbf{;}\mathbf{}\stackrel{¯}{\mathrm{RO}}â‰\dots \stackrel{¯}{\mathrm{SO}}$

a. If you are also given that $\mathbf{m}∠\mathbf{1}\mathbf{=}\mathbf{40}$, find the measures of$∠\mathbf{2}\mathbf{,}\mathbf{}∠\mathbf{7}\mathbf{,}\mathbf{}∠\mathbf{5}$and $∠\mathbf{6}$. Then decide whether$\stackrel{¯}{\mathrm{PQ}}$must be parallel to $\stackrel{¯}{\mathrm{SR}}$.

b. Repeat part (a), but use $\mathbf{m}∠\mathbf{1}\mathbf{=}\mathbf{k}$.

State whether the congruence of triangles have the reflexive property, the symmetric property, the transitive property.

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$x=\frac{−bÂ\pm \sqrt{b^2−4ac}}{2a}$.

$x^2−5x+3=0$

State whether the congruence of triangles has the reflexive property, the symmetric property, the transitive property.

State whether the congruence of triangles have the reflexive property, the symmetric property, the transitive property.

Complete.

a. If$\mathbf{m}∠\mathbf{1}\mathbf{=}\mathbf{20}$ then$\mathbf{m}∠\mathbf{3}\mathbf{=}\mathbf{?},\mathbf{m}∠\mathbf{4}\mathbf{=}\mathbf{?}$ and$\mathbf{m}∠\mathbf{5}\mathbf{=}\mathbf{?}$

b. If$\mathbf{m}∠\mathbf{1}\mathbf{=}\mathbf{x}$ then$\mathbf{m}∠\mathbf{3}\mathbf{=}\mathbf{?},\mathbf{m}∠\mathbf{4}\mathbf{=}\mathbf{?}$ and$\mathbf{m}∠\mathbf{5}\mathbf{=}\mathbf{?}$

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.

In an isosceles triangle, if a segment is drawn from the vertex of the angle between the congruent sides to the midpoint of the opposite side, then congruent triangles are formed.

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.)

23. Given:$\stackrel{↔}{SR}$is the $âŠ\yen$bisector of$\stackrel{¯}{QT};\stackrel{↔}{QR}$is the$âŠ\yen$bisector of$\stackrel{¯}{SP}$

Prove:$PQ=TS$

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.