Question 1
A student writes an incorrect step while checking if the sum of the measures of the two remote interior angles of triangle ABC below is equal to the measure of the exterior angle.
A triangle ABC is shown. The base of triangle extends into a straight line. The angle formed between this straight line and the edge of triangle is marked as d. The angle adjacent to w is marked as c, and the other two angles inside the triangle are marked as a and b.
Step 1: m∠a + m∠b + m∠c = 90 degrees (complementary angles)
Step 2: m∠d + m∠c = 180 degrees (supplementary angles)
Step 3: Therefore, m∠a + m∠b + m∠c = m∠d + m∠c
Step 4: So, m∠a + m∠b = m∠d
In which step did the student first make a mistake and how can it be corrected?
Step 1; it should be m∠a + m∠b + m∠c = 180 degrees (corresponding angles)
Step 1; it should be m∠a + m∠b + m∠c = 180 degrees (sum of angles of a triangle)
Step 2; it should be m∠d − m∠c = 90 degrees (alternate interior angles)
Step 2; it should be m∠d − m∠c = 90 degrees (adjacent angles)
Question 2
The figure shows two similar triangles:
Two triangles are shown. The sides of the triangle on the left are marked 6, 8, 4. The sides of the triangle on the right are marked as 3, 4, and 2. For the triangle on the left, the angle between sides marked 8 and 6 is labeled as v, marked with a double arc, and the angle between the sides marked 8 and 4 is labeled as w, marked with a single arc. The third angle is marked with a triple arc. For the triangle on the right, the angle between sides marked 2 and 4 is labeled as x, marked with a single arc and the angle between the sides marked 4 and 3 is labeled as y, marked with a double arc. The angle between the sides 2 and 3 is labeled as z, marked with a triple arc, and is also the angle on the top vertex of this triangle.
Which two sets of angles are corresponding angles?
∠v and ∠x; ∠w and ∠y
∠v and ∠y; ∠w and ∠x
∠v and ∠z; ∠w and ∠x
∠v and ∠z; ∠w and ∠y
Question 3
The figure below shows two parallel lines cut by a transversal:
A pair of parallel lines is shown with arrowheads on each end. A transversal cuts through these two lines. An angle formed between the top parallel line and the transversal on the outer right side is marked 1. Another angle formed between the top parallel line, and the transversal on the inner right side is marked 2.
Which statement is true about ∠1 and ∠2?
∠1 and ∠2 are complementary, because they are a pair of adjacent angles.
∠1 and ∠2 are supplementary, because they are pair of adjacent angles.
∠1 and ∠2 are complementary, because they are a pair of corresponding angles.
∠1 and ∠2 are supplementary, because they are pair of corresponding angles.
Question 1: The student first makes a mistake in Step 1. It should be m∠a + m∠b + m∠c = 180 degrees (sum of angles of a triangle). This step is corrected by changing the equation to m∠a + m∠b + m∠c = 180 degrees.
Question 2: The two sets of corresponding angles are ∠v and ∠y; ∠w and ∠x. These angles correspond to each other in the two similar triangles.
Question 3: ∠1 and ∠2 are supplementary angles because they are a pair of adjacent angles. When a transversal cuts through parallel lines, the alternate interior angles formed by the transversal and the parallel lines are supplementary.
Question 1
The student made a mistake in Step 1. It should be m∠a + m∠b + m∠c = 180 degrees (sum of angles of a triangle).
Step 1; it should be m∠a + m∠b + m∠c = 180 degrees (sum of angles of a triangle)
Question 2
The corresponding angles in the two triangles are ∠v and ∠x, and ∠w and ∠y.
∠v and ∠x; ∠w and ∠y
Question 3
∠1 and ∠2 are supplementary angles because they are a pair of corresponding angles.
∠1 and ∠2 are supplementary, because they are pair of corresponding angles.
#1. Step 1. The angles add up to what?
#2. The triangles are clearly similar. So draw them in the same orientation (that is, with the triple arc on top). It should be immediately clear which angles and sides correspond.
#3. you're kidding, right? What does supplementary mean? Look at the figure!