Practice: transformations and congruent triangles

Geometry, grade 8 Lesson 12 of 21Practice

Practice: transformations and congruent triangles

Write the coordinates of $\displaystyle P'$ , $\displaystyle Q'$ and $\displaystyle R'$ , the images of the vertices of triangle $\displaystyle PQR$ .

A translation 4 squares to the right and 2 squares down.

A reflection in the $\displaystyle x$ -axis.

A rotation about the origin by $\displaystyle 180^{\circ}$ .

Given: $\displaystyle \triangle PQR\cong\triangle XYZ$ .
$\displaystyle {PQ}={4}$ , $\displaystyle {QR}={6}$ .
$\displaystyle {\angle P}={35^{\circ}}$ , $\displaystyle {\angle Z}={95^{\circ}}$ .
Find $\displaystyle XY$ , $\displaystyle YZ$ , and all the angles of both triangles.

Are triangles $\displaystyle ABC$ and $\displaystyle DEF$ always congruent?
If so, write the congruence theorem. If not, give a reason.

$\displaystyle {AB}={DE}$ , $\displaystyle {\angle A}={\angle D}$ , $\displaystyle {AC}={DF}$ .

$\displaystyle {\angle A}={\angle D}$ , $\displaystyle {AB}={DE}$ , $\displaystyle {\angle B}={\angle E}$ .

$\displaystyle {AB}={DE}={10}$ , $\displaystyle {BC}={EF}={6}$ , $\displaystyle {\angle C}={\angle F}={50^{\circ}}$ .

$\displaystyle {AB}={DE}={4}$ , $\displaystyle {BC}={EF}={9}$ , $\displaystyle {\angle C}={\angle F}={20^{\circ}}$ .

In the quadrilateral $\displaystyle ABCD$ : $\displaystyle {AB}={CD}$ , and $\displaystyle {AD}={BC}$ .
$\displaystyle BD$ is a diagonal.
Prove: $\displaystyle {\angle BAD}={\angle DCB}$ .

In the quadrilateral $\displaystyle ABCD$ , the diagonal $\displaystyle AC$ bisects the angle $\displaystyle A$ and the angle $\displaystyle C$ :
$\displaystyle {\angle BAC}={\angle DAC}$ , and $\displaystyle {\angle BCA}={\angle DCA}$ .
Prove: $\displaystyle {AB}={AD}$ .

$\displaystyle AB\parallel CD$ , and $\displaystyle {AB}={CD}$ .
The segments $\displaystyle AC$ and $\displaystyle BD$ cross at the point $\displaystyle O$ .
Prove: $\displaystyle {AO}={CO}$ .

Given: $\displaystyle \triangle ABC\cong\triangle DEF$ .

$\displaystyle {AB}={2x+1}$ , and $\displaystyle {DE}={x+5}$ .
Write an equation and find $\displaystyle x$ . How long is $\displaystyle AB$ ?

\displaystyle AB

/ – 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

$\displaystyle {\angle C}={(3y)^{\circ}}$ , and $\displaystyle {\angle F}={(y+40)^{\circ}}$ .
Write an equation and find $\displaystyle y$ . How large is the angle $\displaystyle C$ ?

\displaystyle \angle C

/ – 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

Formulas: geometry, grade 8

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Angles between parallel lines

Adjacent angles
$\displaystyle \alpha+\beta=180^{\circ}$
Vertical angles are equal
$\displaystyle \alpha=\gamma$
Between parallel lines: corresponding angles are equal, and so are alternate angles
$\displaystyle \alpha=\beta$
Between parallel lines: co-interior angles
$\displaystyle \gamma+\delta=180^{\circ}$

The angles of a triangle

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
$\displaystyle \delta$ $\displaystyle =$ $\displaystyle \alpha+\beta$
In an isosceles triangle the base angles are equal
$\displaystyle \angle B=\angle C$ $\displaystyle AB=AC$
In an equilateral triangle all the angles are equal
$\displaystyle \alpha=\beta=\gamma=60^{\circ}$

Pythagoras' theorem

In a right triangle – $\displaystyle a$ , $\displaystyle b$ the legs, $\displaystyle c$ the hypotenuse
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
Finding a leg – from the hypotenuse and the other leg
$\displaystyle a^2$ $\displaystyle =$ $\displaystyle c^2-b^2$

The sides of a triangle

The triangle inequality – two sides together are longer than the third
$\displaystyle a+b>c$
The third side – between the difference of the other two and their sum
$\displaystyle a-b<c<a+b$ $\displaystyle a\ge b$

Similar triangles

In similar triangles $\displaystyle \triangle ABC\sim\triangle DEF$ the corresponding sides are in one ratio – $\displaystyle k$ the ratio of similarity
$\displaystyle \frac{DE}{AB}=\frac{EF}{BC}=\frac{FD}{CA}=k$
The ratio of the perimeters – the perimeter grows $\displaystyle k$ times too
$\displaystyle P_{DEF}$ $\displaystyle =$ $\displaystyle k\cdot P_{ABC}$
Cross-multiplying a proportion
$\displaystyle a\cdot d$ $\displaystyle =$ $\displaystyle b\cdot c$ $\displaystyle \frac{a}{b}=\frac{c}{d}$