Angles and Geometric Constructions: practice solutions, Grade 7 – download the PDF

1 Complement and supplement ★★★

Complement: \(90-38=52\), so \(52^\circ\).

Supplement: \(180-38=142\), so \(142^\circ\).

The complement of \(38^\circ\) is \(52^\circ\) and its supplement is \(142^\circ\).

2 Sorting pairs ★★★

1. \(18+72=90\): complementary.
2. \(110+70=180\): supplementary.
3. \(44+47=91\): neither.
4. \(120+60=180\): supplementary.

3 Crossing lines ★★★

Angle \(b\) is vertical to the \(72^\circ\) angle, so \(b=72^\circ\).

Angle \(a\) forms a linear pair with the \(72^\circ\) angle: \(a=180-72=108^\circ\).

Angle \(c\) is vertical to \(a\), so \(c=108^\circ\).

Answer: \(a=108^\circ\), \(b=72^\circ\), \(c=108^\circ\).

4 Can it be a triangle? ★★★

1. \(5+6=11>10\): yes.
2. \(2+3=5<7\): no.
3. \(8+8=16>15\): yes.
4. \(4+9=13\), which is not greater than 13: no, the triangle would be flat.

5 Name the cross section ★★★

1. A square, the same size as the face.
2. A circle.
3. A circle.
4. A triangle.

6 Angles on a line ★★★

Angles on a straight line add up to \(180^\circ\).

\(40+75+x=180\), so \(115+x=180\) and \(x=65\).

The angle \(x\) measures \(65^\circ\).

7 The leaning ladder ★★★

The wall and the ground meet at \(90^\circ\), so the two angles of the ladder are complementary.

\(90-74=16\).

The ladder makes a \(16^\circ\) angle with the wall.

8 A supplementary equation ★★★

\((x+14)+(2x-5)=180\), so \(3x+9=180\) and \(3x=171\), thus \(x=57\).

The angles are \(57+14=71\) and \(2(57)-5=109\), so \(71^\circ\) and \(109^\circ\).

Check: \(71+109=180\).

9 Vertical angles with variables ★★★

Vertical angles are equal: \(4x-7=2x+21\), so \(2x=28\) and \(x=14\).

The angle is \(4(14)-7=49\). Check: \(2(14)+21=49\).

Each angle measures \(49^\circ\).

10 One angle four times the other ★★★

Let the smaller angle be \(x\). Then \(x+4x=180\), so \(5x=180\) and \(x=36\).

The angles are \(36^\circ\) and \(4(36)=144^\circ\).

11 Twenty degrees more ★★★

Let the smaller angle be \(x\); the larger is \(x+20\). Then \(2x+20=90\), so \(x=35\).

The angles are \(35^\circ\) and \(55^\circ\).

Check: \(35+55=90\) and \(55-35=20\).

12 Range for the third side ★★★

\(s<9+14=23\) and \(s>14-9=5\), so \(5

The whole numbers are \(6, 7, \dots, 22\). That is \(22-6+1=17\) values.

13 Craft sticks ★★★

The third length must satisfy \(12-7

4 cm: no. 6 cm: yes. 18 cm: yes. 19 cm: no, because \(12+7=19\) is not greater than 19.

Dana can use 6 cm or 18 cm.

14 How many triangles? ★★★

1. \(5+5=10<12\): no triangle.
2. \(6+8=14>10\): exactly one triangle.
3. The angles add up to \(180^\circ\) but no length is given: many triangles of different sizes.
4. Two sides and the included angle: exactly one triangle.

15 Slicing a box ★★★

A cut parallel to a face gives a rectangle congruent to that face.

1. \(10\times 6=60\text{ in}^2\).
2. \(6\times 4=24\text{ in}^2\).

16 A linear pair in a crossing ★★★

The angles form a linear pair: \((2x+10)+(3x+20)=180\), so \(5x+30=180\) and \(x=30\).

The angles are \(2(30)+10=70\) and \(3(30)+20=110\). Check: \(70+110=180\).

The vertical angle of the first one is also \(70^\circ\).

17 Three angles on a line ★★★

The sum is \(180\): \((x+10)+(2x-15)+(3x+5)=180\), so \(6x=180\) and \(x=30\).

The angles are \(40^\circ\), \(2(30)-15=45^\circ\) and \(3(30)+5=95^\circ\).

Check: \(40+45+95=180\).

18 Complement and supplement together ★★★

Let the angle be \(a\). Then \(90-a=\dfrac{180-a}{4}\). Multiply by 4: \(360-4a=180-a\), so \(180=3a\) and \(a=60\).

Check: complement \(30^\circ\), supplement \(120^\circ\), and \(120\div 4=30\).

The angle measures \(60^\circ\).

19 Three times the complement ★★★

\(180-a=3(90-a)\), so \(180-a=270-3a\) and \(2a=90\), thus \(a=45\).

Check: supplement \(135^\circ\), complement \(45^\circ\), and \(3\times 45=135\).

The angle measures \(45^\circ\).

20 Sides with a variable ★★★

\(x+(x+3)>2x+1\) gives \(2x+3>2x+1\), which is always true.

\(x+(2x+1)>x+3\) gives \(3x+1>x+3\), so \(2x>2\) and \(x>1\).

\((x+3)+(2x+1)>x\) gives \(3x+4>x\), true for positive \(x\).

So \(x>1\); the smallest whole number is \(x=2\), with sides 2 cm, 5 cm and 5 cm. Check: \(2+5>5\).

21 Construction plan ★★★

1. \(180-50-60=70\), so the third angle is \(70^\circ\).
2. Draw \(AB=8\text{ cm}\). With the protractor centered on \(A\), mark \(50^\circ\) and draw a ray. Center it on \(B\), mark \(60^\circ\) from side \(BA\) and draw a ray. Point \(C\) is where the rays cross.
3. Yes: two angles and the included side determine exactly one triangle.

22 Slicing a cone ★★★

Halfway up, the radius is \(5\div 2=2.5\text{ cm}\).

Area \(=\pi r^2=\pi(2.5)^2=6.25\pi\approx 19.6\text{ cm}^2\).

The cross section is a circle of radius 2.5 cm and area about 19.6 cm².

23 The garden fence ★★★

Triangle inequality: \(15-8

Fence limit: \(15+8+s\le 40\), so \(s\le 17\).

Therefore \(8\le s\le 17\), which gives \(17-8+1=10\) possible lengths.