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Angles in Shapes

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Angles in Shapes
CCSS.MATH.CONTENT.4.MD.C.5

Basics on the topic Angles in Shapes

Content

In this Angles in Shapes Video

Nico and Nia are at the amusement park and have time to ride one more ride. They want to choose the roller coaster with the biggest drop. We can help Nico and Nia figure out which ride to choose by looking at the inside angles created by the roller coaster rides.

What are Angles in Shapes?

Angles in shapes are created by the interior angles of a polygon. Interior angles are angles created on the inside corners of shapes. All geometric shapes have angles based on their number of sides.

The interior angles of a triangle are made up of three angles. The sum of interior angles of a triangle will ALWAYS measure one hundred eighty degrees.

Quadrilaterals have four angles and they will ALWAYS total three hundred and sixty degrees.

When measuring angles in shapes, we can find the angles by finding the sum of the interior angles given and subtract that number from the total degrees in the shape or use what you know about a shape's angles to find the missing degrees.

Examples of Angles in Two Dimensional Shapes

Element_8.svg

Element_7.svg

Follow up Activities for Finding the Sum of Interior Angles of a Polygon

Following the video is additional practice with finding the measurements of all interior angles. There are exercises and angles in 2d shapes worksheets.

Transcript Angles in Shapes

“The park is closing soon and we have time for ONE more ride, Nia.” “Let’s go on one of the roller coasters!” "Hmmm(...)WHICH ONE?" “ I want to go on the one with the biggest drop!” We can help Nico and Nia figure out which ride to choose by looking at the inside angles created by the roller coaster rides. "Angles in Shapes" All geometric shapes have angles based on their number of sides. A triangle is made up of three angles(...) and when we add up their measurements, it will ALWAYS equal one hundred eighty degrees. Quadrilaterals have FOUR angles and they will ALWAYS total three hundred and sixty degrees. When we look at the outlines of the roller coaster rides, we can see that they make geometric shapes. We can use what we know about angle measurements in shapes to find the measurement of the missing angle at the top of each ride. Let's look at the Obtuse Oblivion. What do you observe about the shape of this ride? The ride is shaped like an obtuse triangle. One angle measures forty degrees, and the other measures thirty degrees. Let’s think about what we know. We know the measure of two angles and the total triangle will measure one hundred eighty degrees. First, solve this by adding the known angles, then SUBTRACTING the sum from one hundred eighty. Forty-plus thirty equals seventy. One hundred eighty minus seventy equals one hundred ten. The missing angle measures one hundred ten degrees. Next, we’ll check out the Right Wrecker! This one looks intense! What do you notice about the shape and angles created by this ride? This shape is a quadrilateral, which means there are four angles and they measure three hundred sixty degrees in total. We are given the angle measurements for THREE of the angles. We have one hundred ten degrees... and thirty. Even though there isn’t a number here, from the box, we know this one is a right angle. What is the measurement of a right angle? It is ninety degrees. To solve for this missing angle, add the known angles and subtract the sum from three hundred sixty. First, we'll add one hundred ten, and ninety and thirty. What is the sum of these angles? The sum of these angles is two hundred, thirty degrees. Now, subtract the sum from three hundred sixty. What is the measure of the missing angle? One hundred thirty degrees. One more ride to consider(...) the Acute Avalanche! This coaster is made up of TWO shapes. Let’s split them apart and solve for the missing angles in each. What is this first shape? This is a parallelogram, which is a type of quadrilateral. We can solve for the missing angle using what we know about parallelograms. A parallelogram has two sets of opposite angles of equal measure. If THIS angle is one hundred twenty degrees,(...) then THIS angle is also one hundred, twenty degrees. If this angle is sixty degrees... what is the measurement of THIS ONE? The opposite angle would also be sixty degrees. To check our answer, add one hundred twenty, (...) sixty, (...) one hundred twenty, and (...) sixty. These angles total three hundred, sixty degrees. Let's look at the EQUILATERAL triangle. What do you notice about the angles? We aren’t given any of the measurements, but we see they are ALL the SAME size. Since a triangle always has a total of one hundred eighty degrees, we can use that to figure out the measurements of the angles. What number can we add three times to equal one hundred eighty degrees? Sixty! Sixty (...) plus sixty (...) plus sixty equals one hundred eighty. Remember, (...) geometric shapes have angles that have set totals. We can solve for missing inside angle measurements in a couple of ways. One way is to find the sum of angles given and subtract that number from the total degrees in the shape. Or use what you know about a shapes' angles to find the missing degrees. “I think Acute Avalanche looks like it has the BEST drop!” “LET’S GO!!” "Nico, are we there yet?" "Yes, we're at the top!" "HANG ON!!!" "WOOOOOO HOOOOO!"

Angles in Shapes exercise

Would you like to apply the knowledge you’ve learned? You can review and practice it with the tasks for the video Angles in Shapes.
  • What are the missing angles?

    Hints

    Remember, this box in the bottom right corner means there is a right-angle, which is always 90°.

    Remember, all of the internal angles of a triangle equal 180°.

    Remember, all of the internal angles of a quadrilateral equal 360°.

    Remember, opposite angles in parallelograms are equal.

    Solution

    Here are the missing angles:

    • The parallelogram had two missing angles of 75° each.
    • The triangle had one missing angle of 90° and one missing angle of 65°.
    ________________________________________________________

    Parallelogram

    • The first part of the roller coaster was made of a parallelogram.
    • We know that parallelograms are quadrilaterals, so opposite angles are equal and all internal angles add up to 360°.
    • We are given two angles of 105°. 105 + 105 = 210.
    • We then subtract 210 from 360, which equals 150.
    • We then divide 150 by 2, to get 75.
    • Therefore each of the missing angles in the parallelogram are 75°.
    Triangle
    • We know that all internal angles of a triangle add up to 180°.
    • We are given one angle of 25°.
    • We can then see a square box in the corner of the triangle meaning it is a right-angle. Right-angles always equal 90°.
    • We then add 25 + 90 = 115.
    • We then subtract 115 from 180, which equals 65.
    • The missing angle is 65°.
  • What are the missing angles?

    Hints

    Remember, the internal angles of a triangle add up to 180°.

    Remember, the internal angles of a quadrilateral add up to 360°.

    Remember, in a square or parallelogram, opposite angles are equal.

    Solution

    Here are the missing angles for each shape:

    Square

    • A square is a quadrilateral, therefore all internal angles must add up to 360°.
    • All internal angles of a square are also right-angles, meaning that the missing angle must be 90°.
    • 90° + 90° + 90° + 90° = 360°.
    Scalene triangle
    • All angles are different in a scalene triangle but all internal angles still add up to 180°.
    • 80° + 40° = 120°.
    • 180° - 120° = 60°.
    • The missing angle is therefore 60°.
    Parallelogram
    • A parallelogram is a quadrilateral, therefore all internal angles add up to 360°.
    • We also know that opposite angles are equal.
    • 60° + 60° = 120°.
    • 360° - 120° = 240°.
    • 240° $\div$ 2 = 120°.
    • The missing angle is therefore 120°.
    Isosceles triangle
    • An isosceles triangle has two equal angles and all internal angles add up to 180°.
    • We were given the two equal angles of 80°.
    • 80° + 80° = 160°
    • 180° - 160° = 20°
    • The missing angle is 20°.

  • Find the missing angles.

    Hints

    Remember, all right-angles are 90°.

    Add up the angles given in the triangles and parallelograms and figure out what is missing.

    Remember:

    • all internal angles in a triangle add up to 180°.
    • all internal angles in a parallelogram (a quadrilateral) add up to 360°.

    Solution

    Here are the correctly highlighted angles:

    Ladder

    • The shape made here is a rectangle.
    • All angles are right-angles, so are all 90° and are highlighted in green.
    Rafters
    • Here is an equilateral triangle.
    • 60° + 60° = 120°
    • 180° - 120° = 60°
    • The missing angle is 60° and is highlighted in blue.
    Cheese picture
    • Here is another equilateral triangle.
    • 60° + 60° = 120°
    • 180° - 120° = 60°
    • The missing angle is 60° and is highlighted in blue.
    Rectangular picture frame
    • The shape here is a rectangle.
    • All angles are right-angles so are all 90° and are highlighted in green.
    Parallelogram picture frame
    • Opposite angles are equal in a parallelogram.
    • 150° + 150° = 300°.
    • 360° - 300° = 60°.
    • 60° $\div$ 2 = 30°
    • The missing angles are therefore both 30° and are highlighted in yellow.
    Bookcase
    • Here we have a right-angled triangle.
    • One missing angle is a right-angle and is 90° so is highlighted in green.
    • 60° + 90° = 150°.
    • 180° - 150° = 30°.
    • The top angle is therefore 30° and is highlighted in yellow.

  • Can you find the mistakes?

    Hints

    Do any of the shapes have right-angles?

    Remember, in parallelograms:

    • All internal angles add up to 360°
    • Opposite angles are equal

    All internal angles in a triangle add up to 180°.

    Solution

    Here are the angles that should have been highlighted:

    • In the scalene triangle we were given two angles of 48° and 62°. 48 + 62 = 110. 180 - 110 = 70 so the angle was 70° and this is correct.
    • In the parallelogram we were given two angles of 125°. 125 + 125 = 250. 360 - 250 = 110. 110 $\div$ 2 = 55 so the angles should have been 55°. The two 45° angles are incorrect.
    • In the isosceles triangle we were given two angles of 73°. 73 + 73 = 146. 180 - 146 = 34 so the angle is 34°. 33° is incorrect.
    • In the right-angled triangle we were given an angle of 15° and a right-angle which is 90°. 90 + 15 = 105. 180 - 105 = 75 so the angle is 75°. 65° is incorrect.
  • What is the missing angle?

    Hints

    Can you remember what all internal angles of a triangle add up to?

    The internal angles of a triangle always add up to 180°.

    If you add the angles given, what is the difference between that and 180°?

    Solution

    The missing angle was 35°.

    • All internal angles of a triangle add up to 180°.
    • 80 + 65 = 145.
    • 180 - 145 = 35.
    • Therefore the missing angle is 35°.
  • What are the angles that Nico and Nia saw?

    Hints

    A right-angle is always 90°.

    The internal angles of a quadrilateral add up to 360°.

    The internal angles of a triangle add up to 180°.

    An isosceles triangle has two equal angles.

    Solution

    The first roller coaster that Nia and Nico saw had a triangle shape in it. They measured two angles of 72° and 41° so the final angle must be 67°.

    • 72 + 41 = 113.
    • 180 - 113 = 67
    • The missing angle is therefore 67°.
    They then saw one that had a parallelogram shape in it. Two angles measured 84° each so the other two must measure 96° each.
    • A parallelogram is a quadrilateral so all inside angles add up to 360°.
    • 84 + 84 = 168
    • 360 - 168 = 192
    • 192 $\div$ 2 = 96
    • The missing angles are therefore 96° each.
    They then saw a roller coaster that had a right-angled triangle in it. One angle measured 29° so the missing angle must measure 61°.
    • One angle is a right angle so is therefore 90°.
    • 90 + 29 = 119
    • 180 - 119 = 61
    • Therefore the missing angle is 61°.
    Finally they saw a roller coaster with an isosceles triangle. The top angle measured 18° so the bottom two angles must measure 81° each.
    • Isosceles triangles have two equal angles.
    • 180 - 18 = 162
    • 162 $\div$ 2 = 81
    • Therefore the missing angles are 81° each.