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Today, we're going to learn how to find missing angles in a quadrilateral. Do you all remember the angle sum property?
Yes! The sum of the angles is always 360 degrees!
Correct! So how would we find the fourth angle if we have three angles already?
We add the three angles together and subtract from 360.
Exactly! Let's look at an example. If we have angles of 90Β°, 85Β°, and 95Β°, whatβs the fourth angle?
360 minus (90 + 85 + 95) is 90!
Great job! Therefore, the four angles are 90Β°, 85Β°, 95Β°, and 90Β°. This method always works for any quadrilateral.
So, we use the same method for any quadrilateral?
Thatβs right! Remember: always add the known angles and subtract from 360Β°.
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Next, let's explore properties of angles in a parallelogram. What do you remember about the angles in this shape?
Opposite angles are equal!
Exactly! If one angle measures 70Β°, how can we find the remaining angles?
The opposite angle is also 70Β°, and the adjacent angles would be 110Β°.
Right! So the angles in this parallelogram are 70Β°, 110Β°, 70Β°, and 110Β°. Always remember that adjacent angles in a parallelogram sum up to 180Β°.
Are there any other properties we need to keep in mind for parallelograms?
Good question! Not only do we have equal opposite angles, but the diagonals bisect each other. We will explore more about that in the next class!
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The examples in this section demonstrate the application of various properties of quadrilaterals, including finding angles, understanding the characteristics of parallelograms, and calculating areas using established formulas. These exercises reinforce theoretical knowledge with practical applications.
This section details two primary examples related to quadrilaterals, specifically focusing on angles within quadrilaterals and the properties of parallelograms. Through example problems, we see the application of the angle sum property of quadrilaterals β which states that the sum of interior angles is always 360Β°. The first example shows how to calculate the fourth angle in a quadrilateral when the other three angles are known. The second example delves into finding all angles of a parallelogram given one angle. By illustrating these concepts through practical scenarios, students can better grasp the theoretical properties associated with quadrilaterals and parallelograms.
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Example 1:
Find the fourth angle of a quadrilateral if the other three are 90Β°, 85Β°, and 95Β°.
Sum = 360Β°, So fourth angle = 360 - (90 + 85 + 95) = 90Β°.
In this example, we want to find the fourth angle of a quadrilateral when three angles are given. The total sum of angles in any quadrilateral is always 360 degrees. Therefore, we can calculate the fourth angle by subtracting the sum of the known angles from 360 degrees. We add the three given angles: 90 + 85 + 95 = 270 degrees. Now, we subtract this sum from 360 degrees: 360 - 270 = 90 degrees. So, the fourth angle is 90Β°.
Imagine you are baking a cake and have cut out three slices, representing the three angles of the quadrilateral. If you know the angles of the three slices total to a certain amount (270Β°), you can easily calculate how much of the cake (the fourth angle) is left to complete the whole cake (360Β°).
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Example 2:
In a parallelogram, one angle is 70Β°. Find all angles.
Other angle = 180 - 70 = 110Β°. So angles are: 70Β°, 110Β°, 70Β°, 110Β°.
In a parallelogram, opposite angles are equal, and adjacent angles are supplementary (they add up to 180 degrees). Given that one angle is 70Β°, we can find the angle directly opposite to it, which is also 70Β°. To find the adjacent angles, we subtract 70 from 180: 180 - 70 = 110Β°. Thus, the angles in the parallelogram are 70Β°, 110Β°, 70Β°, and 110Β°.
Think of a parallelogram like a pair of opposing doors. One door is at a 70Β° angle relative to the wall, and its opposite door must match it at 70Β° as well. The adjacent corners, where the doors meet the wall, form a straight line with the doors, summing up to 180Β°. So, if one door is open at 70Β°, the adjacent wall corner must be 110Β°.
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Key Concepts
Quadrilateral: A polygon with four sides, four angles, and four vertices.
Angle Sum: The interior angles of a quadrilateral sum to 360Β°.
Parallelogram Properties: Opposite angles and sides are equal, and the diagonals bisect each other.
See how the concepts apply in real-world scenarios to understand their practical implications.
Example 1: Find the fourth angle of a quadrilateral if the other three are 90Β°, 85Β°, and 95Β°. Answer: 360 - (90 + 85 + 95) = 90Β°.
Example 2: In a parallelogram, if one angle measures 70Β°, the other angles are 70Β°, 110Β°, 70Β°, and 110Β°.
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
Four sides, four angles, they meet at a point, sum them all up, three sixty's the joint!
Imagine a friendly quadrilateral named Qaddy, who always gathers his four angles to sum up to 360 degrees during parties, reminding everyone that angles add up just like fun!
To remember properties of a parallelogram, think 'O P D' - Opposite sides equal, Opposite angles equal, Diagonals bisect each other.
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Review the Definitions for terms.
Term: Quadrilateral
Definition:
A polygon with four sides, four vertices, and four angles.
Term: Angle Sum Property
Definition:
The rule that the sum of the interior angles of a polygon is (n-2) Γ 180Β°, where n is the number of sides.
Term: Parallelogram
Definition:
A quadrilateral with opposite sides parallel and equal.