Deductive Geometry (Reasoning) 1

Years

Year 10

Geometry

Deductive Geometry (Reasoning)

Deductive Geometry (Reasoning) 1

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Question 1 of 4

1. Question
Find the value of $x$
- $x=$ (60) $°$
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The sum of the interior angles in a triangle is 180°

Find the value of the angle alternate to $x$ . Let this angle be $y$ .

First, since the interior angles of a triangle add to $180°,$ add the angle measures and set their sum to $180°.$ Then, solve for $y$ .

Note that the angle marked with a square is a right angle, which is equal to $90°.$

$y+90+30$ $=$ $180$

$y+120$ $=$ $180$ Simplify

$y+120$ $-120$ $=$ $180$ $-120$ Subtract $120$ from both sides

$y$ $=$ $60°$

Finally, we can see from the diagram that $y$ and $x$ are alternate angles, which means they are equal

Therefore, $/_ x=60°$

$/_ x=60°$
Question 2 of 4

2. Question
Find the value of $a$
- $a=$ (73) $°$
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Supplementary angles are when two angles have a sum of $180°.$ Typically, these angles lie on a straight line.

The sum of the interior angles in a triangle is 180°

Find the two missing interior angles in the triangle then solve for $a$ by equating them to $180°$

Let the first missing interior angle be $b$ .
We can see from the diagram that $40°$ and $b$ are alternate angles, which means they are equal

Therefore, $/_ b=40°$

Next, let the second missing interior angle be $c$ .
We can see from the diagram that the exterior angle $113°$ and the interior angle $c$ lie on a straight line. Therefore, they are supplementary angles

Since supplementary angles add to $180°,$ add the angle measures and set their sum to $180°.$ Then, solve for the value of $c$ .

$c+113$ $=$ $180$

$c+113$ $-113$ $=$ $180$ $-113$ Subtract $113$ from both sides

$c$ $=$ $67°$

Finally, since the interior angles of a triangle add to $180°,$ add the angle measures and set their sum to $180°.$ Then, solve for $a$ .

$a+b+c$ $=$ $180$

$a+40+67$ $=$ $180$ Plug in the known values

$a+107$ $=$ $180$ Simplify

$a+107$ $-107$ $=$ $180$ $-107$ Subtract $107$ from both sides

$a$ $=$ $73°$

$/_ a=73°$

Question 3 of 4

Find the value of

$a$ and

$b$

$a=$ (110) $°$

$b=$ (35) $°$

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An Isosceles Triangle has two congruent sides (the two sides with dashes) and the two base angles are equal.

The sum of the interior angles in a triangle is 180°

To find the value of

$b$ , first find the value of

$/_KML$

To find the value of

$a$ , first find the two missing interior angles in the triangle then solve for

$a$ by equating them to

$180°$

First, since the base angles in an isosceles triangle are equal, $/_MKL$ and $/_KML$ are equal

Let their value be

$x$

Next, since the interior angles of a triangle add to

$180°,$ add the angle measures and set their sum to

$180°.$ Then, solve for

$x$ .

$x+x+110$	$=$	$180$
$2x+110$	$=$	$180$	Simplify
$2x+110$ $-110$	$=$	$180$ $-110$	Subtract $110$ from both sides
$2x$	$=$	$70$
$2x$ $divide2$	$=$	$70$ $divide2$	Divide both sides by $2$
$x$	$=$	$35°$

Next, we can see from the diagram that $110°$ and $35°+b$ are co-interior angles, which add up to

$180°$

Since co-interior angles add to

$180°,$ add the angle measures and set their sum to

$180°.$ Then, solve for the value of

$b$ .

$b+x+110$	$=$	$180$
$b+35+110$	$=$	$180$	Plug in the known values
$b+145$	$=$	$180$	Simplify
$b+145$ $-145$	$=$	$180$ $-145$	Subtract $145$ from both sides
$b$	$=$	$35°$

Now, since the base angles in an isosceles triangle are equal, $/_KJM$ is also equal to $35˚$

$/_KJM$

$=$

$35$

Finally, since the interior angles of a triangle add to

$180°,$ add the angle measures and set their sum to

$180°.$ Then, solve for

$a$ .

$a+b+/_KJM$	$=$	$180$
$a+35+35$	$=$	$180$	Plug in the known values
$a+70$	$=$	$180$	Simplify
$a+70$ $-70$	$=$	$180$ $-70$	Subtract $70$ from both sides
$a$	$=$	$110°$

$/_ a=110°$

$/_ b=35°$

Question 4 of 4

4. Question
Find the value of $x$ , $y$ and $z$
- $x=$ (70) $°$
  
  $y=$ (70) $°$
  
  $z=$ (40) $°$
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Supplementary angles are when two angles have a sum of $180°.$ Typically, these angles lie on a straight line.

The sum of the interior angles in a triangle is 180°

To find the value of $y$ , first find its alternate angle

To find the value of $x$ , first find its supplementary angle and set their sum to $180°$

To find the value of $z$ , first find the two missing interior angles in the triangle then solve for $z$ by equating them to $180°$

First, we can see from the diagram that $70°$ and $y$ are alternate angles, which means they are equal

Therefore, $y=70°$

Next, we can see from the diagram that the exterior angle $110°$ and the interior angle $x$ lie on a straight line. Therefore, they are supplementary angles

Since supplementary angles add to $180°,$ add the angle measures and set their sum to $180°.$ Then, solve for the value of $x$ .

$x+110$ $=$ $180$

$x+110$ $-110$ $=$ $180$ $-110$ Subtract $110$ from both sides

$x$ $=$ $70°$

Finally, since the interior angles of a triangle add to $180°,$ add the angle measures and set their sum to $180°.$ Then, solve for $z$ .

$z+x+y$ $=$ $180$

$z+70+70$ $=$ $180$ Plug in the known values

$z+140$ $=$ $180$ Simplify

$z+140$ $-140$ $=$ $180$ $-140$ Subtract $140$ from both sides

$z$ $=$ $40°$

$/_ x=70°$

$/_ y=70°$

$/_ z=40°$

Deductive Geometry (Reasoning) 1

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2. Question

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4. Question

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Quizzes

$y+90+30$	$=$	$180$
$y+120$	$=$	$180$	Simplify
$y+120$ $-120$	$=$	$180$ $-120$	Subtract $120$ from both sides
$y$	$=$	$60°$

$c+113$	$=$	$180$
$c+113$ $-113$	$=$	$180$ $-113$	Subtract $113$ from both sides
$c$	$=$	$67°$

$a+b+c$	$=$	$180$
$a+40+67$	$=$	$180$	Plug in the known values
$a+107$	$=$	$180$	Simplify
$a+107$ $-107$	$=$	$180$ $-107$	Subtract $107$ from both sides
$a$	$=$	$73°$

$x+110$	$=$	$180$
$x+110$ $-110$	$=$	$180$ $-110$	Subtract $110$ from both sides
$x$	$=$	$70°$

$z+x+y$	$=$	$180$
$z+70+70$	$=$	$180$	Plug in the known values
$z+140$	$=$	$180$	Simplify
$z+140$ $-140$	$=$	$180$ $-140$	Subtract $140$ from both sides
$z$	$=$	$40°$