Points, lines and angles

Unit 6 Points, lines and angles

Key unit competence: Explore properties of angles formed by lines.

Key ideas

• A point marks a position; a line goes on forever in both directions; a segment $[AB]$ has two end points; a ray has one end point and goes on in one direction.
• Angle types: acute ($<90^{\circ }$), right ($=90^{\circ }$), obtuse (between $90^{\circ }$ and $180^{\circ }$), straight ($=180^{\circ }$), reflex (between $180^{\circ }$ and $360^{\circ }$).
• Angles on a straight line add up to $180^{\circ }$; angles at a point add up to $360^{\circ }$; vertically opposite angles are equal.
• Complementary angles add up to $90^{\circ }$; supplementary angles add up to $180^{\circ }$.
• The interior angles of a polygon with $n$ sides add up to $(n-2)\times 180^{\circ }$. The exterior angles of any convex polygon add up to $360^{\circ }$. In a regular polygon each exterior angle is $\frac{360^{\circ }}{n}$.
• When a transversal cuts parallel lines: corresponding angles are equal, alternate angles are equal, and co-interior angles add up to $180^{\circ }$.

Formulas to know

Common misconceptions

• "Co-interior angles are equal." No: they are supplementary (sum $180^{\circ }$); alternate and corresponding angles are the equal ones.
• "A line has two end points." No: a segment has two end points; a line goes on forever.
• "The angle sum of a polygon is $n\times 180^{\circ }$." No: it is $(n-2)\times 180^{\circ }$, because the polygon splits into $n-2$ triangles.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. Name the type of each angle: $89^{\circ }$, $90^{\circ }$, $91^{\circ }$, $180^{\circ }$, $270^{\circ }$.

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   acute, right, obtuse, straight, reflex.

   Common mistake: Calling $180^{\circ }$ obtuse.

2. Explain why the angles of any quadrilateral add up to $360^{\circ }$.

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   One diagonal splits the quadrilateral into 2 triangles; $2\times 180^{\circ }=360^{\circ }$.

   Common mistake: Drawing both diagonals and counting 4 triangles (the centre angles are then extra).

3. Angles on a straight line are $3x$, $2x+10^{\circ }$ and $x+20^{\circ }$. Find $x$ and each angle.

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   $6x+30=180$, $x=25^{\circ }$; angles $75^{\circ }$, $60^{\circ }$, $45^{\circ }$.

   Common mistake: Using $360^{\circ }$ for angles on a line.

4. In triangle ABC, angle A = $50^{\circ }$ and angle B = $65^{\circ }$. Find angle C and the exterior angle at C.

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   $C=180^{\circ }-115^{\circ }=65^{\circ }$; exterior angle at C $=115^{\circ }$ (equal to A + B).

   Common mistake: Forgetting that the exterior angle equals the sum of the two opposite interior angles.

5. Find the interior angle sum of a heptagon (7 sides) and each angle if it is regular.

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   $(7-2)\times 180^{\circ }=900^{\circ }$; each $\frac{900^{\circ }}{7}\approx 128.6^{\circ }$.

   Common mistake: Using $7\times 180^{\circ }$.

6. Find the angle between the hands of a clock at 3:00 and at 6:00.

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   At 3:00: $90^{\circ }$; at 6:00: $180^{\circ }$ (each hour mark is $30^{\circ }$).

   Common mistake: Using $6^{\circ }$ per mark (that is for minutes).

7. Two parallel lines are cut by a transversal. One of the eight angles is $72^{\circ }$. Find all the others.

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   Four angles are $72^{\circ }$ (vertically opposite, corresponding, alternate) and four are $108^{\circ }$ (supplements).

   Common mistake: Mixing up which pairs are equal and which are supplementary.

8. Is it possible to have a regular polygon whose exterior angle is $50^{\circ }$? Explain.

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   No: $n=\frac{360}{50}=7.2$ is not a whole number.

   Common mistake: Accepting a non-whole number of sides.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Find the angle between the hands of a clock at 2:30.	a. 9
   2. Each interior angle of a regular polygon is $140^{\circ }$. How many sides does it have?	b. $48^{\circ }$
   3. Two lines cross. One angle is $48^{\circ }$. What is the vertically opposite angle?	c. $80^{\circ }$
   4. Three angles of a quadrilateral are $80^{\circ }$, $95^{\circ }$ and $105^{\circ }$. Find the fourth.	d. $105^{\circ }$

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   1 → d ($105^{\circ }$) · 2 → a (9) · 3 → b ($48^{\circ }$) · 4 → c ($80^{\circ }$)

   • Step 1. Minute hand at 30 min: $180^{\circ }$. Hour hand: $2.5\times 30^{\circ }=75^{\circ }$.
     Step 2. Difference: $180^{\circ }-75^{\circ }=105^{\circ }$.
   • Step 1. Each exterior angle $=180^{\circ }-140^{\circ }=40^{\circ }$.
     Step 2. Number of sides $=\frac{360^{\circ }}{40^{\circ }}=9$.
   • Vertically opposite angles are equal.
   • $360^{\circ }-280^{\circ }=80^{\circ }$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $60^{\circ }$  ·  $120^{\circ }$  ·  equal  ·  supplementary
   1. Each interior angle of a regular hexagon measures .
   2. A transversal cuts two parallel lines. Co-interior angles are .
   3. A transversal cuts two parallel lines. Alternate angles are .
   Show answer

   1: $120^{\circ }$ · 2: supplementary · 3: equal

   • $\frac{(6-2)\times 180^{\circ }}{6}=120^{\circ }$.
   • They lie on the same side, between the lines, and add up to $180^{\circ }$.
   • Alternate angles lie on opposite sides of the transversal, between the lines, and are equal.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. What is the sum of the interior angles of a quadrilateral?	a. $360^{\circ }$
   2. A transversal cuts two parallel lines. Two co-interior angles are $(2x+10{)}^{\circ }$ and $(3x-30{)}^{\circ }$. Find $x$.	b. $60^{\circ }$
   3. What is the sum of the interior angles of a pentagon?	c. 40
   4. An angle is twice its complement. Find the angle.	d. $540^{\circ }$

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   1 → a ($360^{\circ }$) · 2 → c (40) · 3 → d ($540^{\circ }$) · 4 → b ($60^{\circ }$)

   • $(4-2)\times 180^{\circ }=360^{\circ }$.
   • Step 1. Co-interior angles add up to $180^{\circ }$: $2x+10+3x-30=180$.
     Step 2. $5x-20=180$, so $5x=200$ and $x=40$.
   • $(5-2)\times 180^{\circ }=540^{\circ }$.
   • Step 1. $x=2(90^{\circ }-x)$ gives $3x=180^{\circ }$.
     Step 2. $x=60^{\circ }$.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $45^{\circ }$  ·  $540^{\circ }$  ·  equal  ·  $360^{\circ }$
   1. A transversal cuts two parallel lines. Corresponding angles are .
   2. The exterior angles of any convex polygon add up to .
   3. Each exterior angle of a regular octagon is .
   Show answer

   1: equal · 2: $360^{\circ }$ · 3: $45^{\circ }$

   • Corresponding angles are in matching positions and are equal for parallel lines.
   • Walking around a polygon you turn through one full turn.
   • $\frac{360^{\circ }}{8}=45^{\circ }$.

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