Question

unit 1 exam on lessons 1.1 to 1.12 (33 points possible)

1. in the accompanying diagram, (overline{ab}perpoverline{bc}) and (mangle abd = 19^{circ}). find (mangle dbc).

(1) (9.5^{circ})
(2) (19^{circ})
(3) (71^{circ})
(4) (161^{circ})

1. in the accompanying diagram, line (m) is parallel to line (n) and they are cut by transversal (t). if the acute angle shown is (15^{circ}), which of the following statements is true?

(1) (mangle x=15^{circ}) and is an obtuse angle
(2) (mangle x = 15^{circ}) and is an acute angle
(3) (mangle x=165^{circ}) and is an obtuse angle
(4) (mangle x=165^{circ}) and is an acute angle

1. in (\triangle abc), (mangle a = 40^{circ}), (mangle b = 75^{circ}), and (mangle c = 65^{circ}). which list has the sides of (\triangle abc) in order from longest to shortest?

(1) (ac>bc > ab)
(2) (bc>ab > ac)
(3) (ab>ac > bc)
(4) (ac>ab > bc)

Explanation:

Step1: Recall perpendicular - angle property

Since $\overline{AB}\perp\overline{BC}$, $\angle ABC = 90^{\circ}$. And $\angle ABC=\angle ABD+\angle DBC$.

Step2: Solve for $\angle DBC$

Given $\angle ABD = 19^{\circ}$, then $\angle DBC=\angle ABC-\angle ABD=90^{\circ}- 19^{\circ}=71^{\circ}$.

Step1: Recall parallel - line and transversal property

When two parallel lines $m$ and $n$ are cut by a transversal $t$, corresponding angles are equal, alternate - interior angles are equal, and same - side interior angles are supplementary. The given acute angle and $\angle x$ are supplementary (linear - pair or same - side interior angles).

Step2: Calculate the measure of $\angle x$

If the acute angle is $15^{\circ}$, then $\angle x=180^{\circ}-15^{\circ}=165^{\circ}$, and an angle with measure greater than $90^{\circ}$ and less than $180^{\circ}$ is an obtuse angle.

Step1: Recall angle - side relationship in a triangle

In a triangle, the side opposite the largest angle is the longest side, and the side opposite the smallest angle is the shortest side.

Step2: Order the angles

Given $\angle A = 40^{\circ}$, $\angle B=75^{\circ}$, $\angle C = 65^{\circ}$. So, $\angle B>\angle C>\angle A$.

Step3: Order the sides

The side opposite $\angle A$ is $BC$, the side opposite $\angle B$ is $AC$, and the side opposite $\angle C$ is $AB$. So, $AC>BC>AB$.

Answer:

(3) $71^{\circ}$