Determine whether the statement is true or false.
1. The complement of an angle that measures [tex]$63^{\circ}$[/tex] has a measure equal to [tex]$127^{\circ}$[/tex].
2. The supplement of an angle that measures [tex]$100^{\circ}$[/tex] has a measure equal to [tex]$180^{\circ}$[/tex].
3. The measures of vertical angles are congruent.
4. Two angles forming a linear pair must be complementary.
5. Vertical angles are formed by two intersecting lines.
Answer (2)
Statements 1 and 2 are false; statement 3 is true; statements 4 and 5 are false and true, respectively.
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Calculate the complement of 6 3 ∘ : 9 0 ∘ − 6 3 ∘ = 2 7 ∘ . Since 2 7 ∘ = 12 7 ∘ , the statement is false.
Calculate the supplement of 10 0 ∘ : 18 0 ∘ − 10 0 ∘ = 8 0 ∘ . Since 8 0 ∘ = 18 0 ∘ , the statement is false.
Vertical angles are congruent: True.
Linear pairs are supplementary, not complementary: False.
Vertical angles are formed by intersecting lines: True.
1. False, 2. False, 3. True, 4. False, 5. True
Explanation
Analyze each statement Let's analyze each statement to determine if it's true or false.
Statement 1: The complement of an angle that measures 6 3 ∘ has a measure equal to 12 7 ∘ .
The complement of an angle is found by subtracting the angle's measure from 9 0 ∘ . So, the complement of 6 3 ∘ is 9 0 ∘ − 6 3 ∘ = 2 7 ∘ . Since 2 7 ∘ = 12 7 ∘ , this statement is false.
Statement 2: The supplement of an angle that measures 10 0 ∘ has a measure equal to 18 0 ∘ .
The supplement of an angle is found by subtracting the angle's measure from 18 0 ∘ . So, the supplement of 10 0 ∘ is 18 0 ∘ − 10 0 ∘ = 8 0 ∘ . Since 8 0 ∘ = 18 0 ∘ , this statement is false.
Statement 3: The measures of vertical angles are congruent. This statement is true based on the vertical angles theorem, which states that vertical angles (angles opposite each other when two lines intersect) are congruent (have the same measure).
Statement 4: Two angles forming a linear pair must be complementary. A linear pair consists of two adjacent angles that form a straight line, meaning their measures add up to 18 0 ∘ . Complementary angles are two angles whose measures add up to 9 0 ∘ . Therefore, two angles forming a linear pair must be supplementary, not complementary. This statement is false.
Statement 5: Vertical angles are formed by two intersecting lines. This statement is true. Vertical angles are indeed formed by the intersection of two lines, where they are the angles opposite each other at the point of intersection.
Final Answer
The complement of an angle that measures 6 3 ∘ has a measure equal to 12 7 ∘ . False
The supplement of an angle that measures 10 0 ∘ has a measure equal to 18 0 ∘ . False
The measures of vertical angles are congruent. True
Two angles forming a linear pair must be complementary. False
Vertical angles are formed by two intersecting lines. True
Examples
Understanding angle relationships is crucial in many real-world applications, such as architecture and construction. For example, when designing a building, architects need to ensure that walls are perpendicular (forming right angles) and that roofs are angled correctly to allow for proper drainage. Similarly, in navigation, understanding angles is essential for determining direction and calculating distances. For instance, pilots use angles to adjust their flight path and avoid obstacles, while sailors use angles to navigate using a compass and charts.
