Flashcards: Angles, triangles & polygons

Rule

Angles on a straight line add up to what?

Answer

$180^\circ$ .

On a straight linea + b = 180°

From the lesson: Angles and parallel lines

Rule

Angles at a point add up to what?

Answer

$360^\circ$ .

At a pointa + b + c = 360°

From the lesson: Angles and parallel lines

Rule

Vertically opposite angles are…?

Answer

Equal.

Vertically oppositeThe two a's are equal, and so are the two b's

From the lesson: Angles and parallel lines

Rule

Corresponding angles on parallel lines are…?

Answer

Equal (they sit in an F shape).

CorrespondingEqual

From the lesson: Angles and parallel lines

Rule

Alternate angles on parallel lines are…?

Answer

Equal (they sit in a Z shape).

AlternateEqual

From the lesson: Angles and parallel lines

Rule

Co-interior angles on parallel lines…?

Answer

Add up to $180^\circ$ (a C shape). They are not equal.

Co-interiora + b = 180°

From the lesson: Angles and parallel lines

Rule

A bend between two parallel lines: how do you find the angle at the corner?

Answer

Draw a third parallel through the corner. The corner angle splits into two alternate angles: it is $a + b$ .

A bend between parallelsDraw a parallel through the corner: the corner angle is a + b

From the lesson: Angles and parallel lines

Rule

The angles of a triangle add up to what?

Answer

$180^\circ$ .

Angle suma + b + c = 180°

From the lesson: Angles in triangles

Rule

The exterior angle of a triangle equals…?

Answer

The sum of the two interior angles opposite it (not the one next to it).

Exterior angleThe exterior angle is a + b

From the lesson: Angles in triangles

Rule

In an isosceles triangle, which angles are equal?

Answer

The two angles facing the equal sides (the base angles).

Isosceles triangleEqual sides (ticks) face equal angles

From the lesson: Angles in triangles

Rule

The interior angles of an

n

-sided polygon add up to what?

Answer

$(n - 2) \times 180^\circ$ . A pentagon: $3 \times 180^\circ = 540^\circ$ .

Interior anglesA pentagon is 3 triangles: (5 − 2) × 180° = 540°

From the lesson: Polygons and quadrilaterals

Rule

The exterior angles of a polygon add up to what?

Answer

$360^\circ$ , for every polygon. For a regular one, $n = \dfrac{360^\circ}{\text{exterior angle}}$ .

Exterior anglesThey add up to 360° for every polygon

From the lesson: Polygons and quadrilaterals

Know it

Each interior angle of a regular polygon is

144^\circ

. How many sides does it have?

Answer

Find the exterior angle first: $180^\circ - 144^\circ = 36^\circ$ . Then $n = 360 \div 36 = 10$ .

From the lesson: Polygons and quadrilaterals

Rule

Pythagoras' theorem?

Answer

$c^2 = a^2 + b^2$ , where $c$ is the hypotenuse, opposite the right angle. For a shorter side, subtract: $a^2 = c^2 - b^2$ .

Pythagorasc is opposite the right angle: c² = a² + b²

From the lesson: Pythagoras, congruent and similar triangles

Rule

The four tests for congruent triangles?

Answer

SSS, SAS (the angle between the two sides), ASA and RHS.

SASTwo sides and the angle between them

From the lesson: Pythagoras, congruent and similar triangles

Rule

Similar triangles: how are the sides and the areas related?

Answer

The angles are the same. Every side is multiplied by the same $k$ , and the area by $k^2$ . Match corners by their equal angles before pairing sides.

Similar trianglesSame angles; every side × k; area × k²

From the lesson: Pythagoras, congruent and similar triangles

Which method?

WAEC 2022 · Paper 2 · Q11 (a)

The exterior angles of a polygon are $42^\circ, 38^\circ, 57^\circ, x^\circ, (x + y)^\circ, (2x - 15)^\circ$ and $(3x - y)^\circ$ . If $x$ is $7^\circ$ less than $y$ , find the values of $x$ and $y$ .

Which fact do you use?

Answer

The exterior angles of any polygon add up to $360^\circ$ . Add them all, set the total to 360, and the $y$ terms cancel.

From the lesson: Polygons and quadrilaterals Try the question

Which method?

WAEC 2014 · Paper 2 · Q3 (a)

In the diagram, $QSP$ and $QTR$ are straight lines, $|PS| = 6\text{ cm}$ , $|QS| = 4\text{ cm}$ , $|QT| = 5\text{ cm}$ and $\angle QTS = \angle RPQ$ . Calculate $|TR|$ .

Which triangles are similar, and how do you pair the sides?

Answer

Triangles $QTS$ and $QPR$ share the angle at $Q$ and have $\angle QTS = \angle QPR$ . So $T$ matches $P$ and $S$ matches $R$ : $\dfrac{QT}{QP} = \dfrac{QS}{QR}$ , using whole sides.

From the lesson: Pythagoras, congruent and similar triangles Try the question