Question

parallel lines and triangles

1. select all true statements if n || m.

a. m∠2 = 60
b. m∠3 = 100
c. m∠2 + m∠4 = 80
d. m∠2 + m∠3 = 80
e. m∠2 = 20

1. select all the true statements.

a. 4x + 16 = 180
b. x = 49
c. m∠a = 99
d. from smallest to largest: ∠b, ∠c, ∠a
e. m∠c = 56

1. line m is constructed as the first step to prove the triangle angle - sum theorem. which of the following must be true in order to complete the proof?

a m || n
b bc = ac
c m∠1 + m∠2 = 90
d m∠1 = m∠2
use the figure shown for items 4 and 5.

1. what is m∠1?

a 130
b 70
c 60
d 120

1. what kind of triangle is bcd? explain.

a right triangle; m∠2 = 90
b acute triangle; m∠cdb = 180 - 130 = 50 and m∠2 = 180 - 70 - 50 = 60
c obtuse triangle; m∠cdb = 180 - 70 = 110
d isosceles triangle; m∠cdb = m∠2

Explanation:

Step1: Analyze question 1

Use properties of parallel - lines and angles in a triangle. If \(n\parallel m\), we use corresponding, alternate - interior, and linear - pair angle relationships. But since no angle measures or relationships are given in the problem statement about the figure for question 1 other than \(n\parallel m\), we assume some basic angle - related facts. However, without the full figure details, we can't fully solve it.

Step2: Analyze question 2

In a triangle, the sum of interior angles is \(180^{\circ}\). So, \((2x + 1)+(x + 15)+y=180\). If we assume the triangle is non - degenerate and we want to find \(x\) and angle measures.
For option A, \(4x+16 = 180\) might come from some angle - addition or linear - pair relationship in the triangle. Solving \(4x+16 = 180\) gives \(4x=180 - 16=164\), then \(x = 41
eq49\) (option B is wrong).
If \(x = 41\), then \(m\angle A=2x + 1=2\times41+1 = 83
eq99\) (option C is wrong).
To order the angles from smallest to largest, we need to find all angle measures. \(m\angle C=x + 15=41+15 = 56\). Let \(m\angle B=y\), then \(m\angle A + m\angle B+m\angle C=180\).

Step3: Analyze question 3

To prove the Triangle Angle - Sum Theorem, we construct a line parallel to one of the sides of the triangle. So, \(m\parallel n\) is a necessary condition to use properties of parallel lines and angle relationships to show that the sum of the interior angles of a triangle is \(180^{\circ}\).

Step4: Analyze question 4

Using the exterior - angle property of a triangle. An exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. If we assume the non - adjacent interior angles to \(\angle1\) are \(70^{\circ}\) and \(60^{\circ}\) (from the figure), then \(m\angle1=70 + 60=130\).

Step5: Analyze question 5

First, find \(m\angle CDB=180 - 130 = 50\). Then, find \(m\angle2=180-(70 + 50)=60\). Since all angles (\(70^{\circ},50^{\circ},60^{\circ}\)) are less than \(90^{\circ}\), it is an acute triangle.

Answer:

1. (No full - solution due to lack of figure details)
2. E. \(m\angle C = 56\) (after calculation as shown above)
3. A. \(m\parallel n\)
4. A. 130
5. B. Acute triangle; \(m\angle CDB = 180 - 130=50\) and \(m\angle2 = 180 - 70 - 50 = 60\)