Practice: triangles and parallel lines

Geometry, grade 9 Lesson 7 of 21Practice

Practice: triangles and parallel lines

1

Answer, and give the reason for each answer.

a

In triangle $\displaystyle ABC$ , $\displaystyle {\angle A}={50^{\circ}}$ and $\displaystyle {\angle B}={60^{\circ}}$ .
Which side is the longest, and which the shortest?

b

In triangle $\displaystyle KLM$ , $\displaystyle {\angle K}={2x,\ \angle L}={3x,\ \angle M}={4x}$ .
Find $\displaystyle x$ , and order the sides from the shortest to the longest.

\displaystyle x

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

In triangle $\displaystyle PQR$ , $\displaystyle {PQ}={5,\ QR}={7,\ PR}={9}$ .
Which angle is the largest, and which the smallest?

2

In triangle $\displaystyle ABC$ , the segment $\displaystyle AM$ is both a median and an altitude: $\displaystyle M$ is the midpoint of $\displaystyle BC$ , and $\displaystyle AM\perp BC$ .
$\displaystyle {AB}={7}$ and $\displaystyle {\angle B}={65^{\circ}}$ .
Find $\displaystyle AC$ and $\displaystyle \angle BAC$ , and give the reasons.

\displaystyle AC

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle \angle BAC

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

3

Calculate, and give the reasons.

a

In triangle $\displaystyle ABC$ , angle $\displaystyle C$ is right, $\displaystyle {\angle A}={30^{\circ}}$ and $\displaystyle {AB}={14}$ .
Find $\displaystyle BC$ and $\displaystyle AC$ .

\displaystyle BC

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle AC

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

In triangle $\displaystyle PQR$ , angle $\displaystyle Q$ is right, $\displaystyle {PR}={10}$ and $\displaystyle {QR}={5}$ .
Find the angles of the triangle.

\displaystyle \angle P

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

\displaystyle \angle R

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

4

In the figure the line $\displaystyle EF$ cuts $\displaystyle AB$ and $\displaystyle CD$ .
In each part: find the $\displaystyle x$ for which $\displaystyle AB\parallel CD$ .

a

$\displaystyle {\angle EPB}={3x+10^{\circ}}$ and $\displaystyle {\angle PQD}={2x+40^{\circ}}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

$\displaystyle {\angle BPQ}={4x}$ and $\displaystyle {\angle PQD}={2x+60^{\circ}}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

$\displaystyle {\angle APQ}={5x-12^{\circ}}$ and $\displaystyle {\angle PQD}={3x+20^{\circ}}$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

5

In triangle $\displaystyle ABC$ , the segment $\displaystyle AD$ is both an altitude and the bisector of angle $\displaystyle A$ .
The point $\displaystyle E$ is on $\displaystyle AD$ .
Prove: $\displaystyle {EB}={EC}$ .

6

In triangle $\displaystyle ABC$ , the segment $\displaystyle BD$ bisects angle $\displaystyle B$ , with the point $\displaystyle D$ on $\displaystyle AC$ .
The point $\displaystyle E$ is on $\displaystyle BC$ , and $\displaystyle {DE}={BE}$ .
Prove: $\displaystyle DE\parallel AB$ .

7

In triangle $\displaystyle ABC$ , angle $\displaystyle C$ is right and $\displaystyle {\angle A}={30^{\circ}}$ .
The point $\displaystyle D$ is on $\displaystyle AC$ , and $\displaystyle BD$ bisects angle $\displaystyle B$ .
Prove: $\displaystyle {AD}={2CD}$ .

8

In triangle $\displaystyle ABC$ , $\displaystyle {AB}={AC}$ .
The point $\displaystyle D$ is on the extension of the side $\displaystyle BC$ , beyond the point $\displaystyle C$ .
Prove: $\displaystyle AD>AB$ .

Formulas: geometry, grade 9

Angles and sides in a triangle

The sum of the angles of a triangle
$\displaystyle \alpha+\beta+\gamma=180^{\circ}$
An exterior angle $\displaystyle \delta$ equals the sum of the two interior angles not next to it
$\displaystyle \delta$ $\displaystyle =$ $\displaystyle \alpha+\beta$
The larger side faces the larger angle, and the other way round
$\displaystyle a>b\ \Leftrightarrow\ \alpha>\beta$

The right triangle

Pythagoras’ theorem – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle a^{2}+b^{2}$ $\displaystyle =$ $\displaystyle c^{2}$
The leg $\displaystyle a$ facing a $\displaystyle 30^{\circ}$ angle is half the hypotenuse $\displaystyle c$
$\displaystyle a$ $\displaystyle =$ $\displaystyle \frac{1}{2}c$
The median $\displaystyle m_c$ to the hypotenuse is half the hypotenuse $\displaystyle c$
$\displaystyle m_c$ $\displaystyle =$ $\displaystyle \frac{1}{2}c$

Angles between parallel lines

Corresponding angles are equal, and so are alternate angles
$\displaystyle \alpha=\beta$
Co-interior angles
$\displaystyle \alpha+\beta=180^{\circ}$

Angles in a quadrilateral

The sum of the angles of a quadrilateral
$\displaystyle \alpha+\beta+\gamma+\delta=360^{\circ}$
In a trapezoid: the two angles beside a leg
$\displaystyle \alpha+\beta=180^{\circ}$

Areas

The area of a triangle – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a\cdot h_a}{2}$
The area of a parallelogram – $\displaystyle h_a$ the height to the side $\displaystyle a$
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot h_a$
The area of a rectangle
$\displaystyle S$ $\displaystyle =$ $\displaystyle a\cdot b$
The area of a square
$\displaystyle S$ $\displaystyle =$ $\displaystyle a^{2}$
The area of a trapezoid – $\displaystyle a$ , $\displaystyle b$ the bases, $\displaystyle h$ the height
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{(a+b)\cdot h}{2}$
The area of a rhombus or a kite – $\displaystyle d_1$ , $\displaystyle d_2$ the diagonals
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{d_1\cdot d_2}{2}$
The ratio of the areas of similar triangles – $\displaystyle k$ the ratio of similarity
$\displaystyle \frac{S_1}{S_2}$ $\displaystyle =$ $\displaystyle k^{2}$