Questions 1-3
A triangle is:
either scalene, isoceles, or equilateral, and
either acute, right, or obtuse.
Question 1: The triangle is right b/c it has a right angle.
The triangle is isoceles because it has two congruent sides.
Right isoceles triangle
Question 2: The triangle is obtuse b/c it has an obtuse angle. (>90°)
The triangle is isoceles b/c it has two congruent sides.
Obtuse isoceles triangle
Question 3: All angles in a triangle add up to 180°. Let's find the 3rd angle.
180 - (80 + 60) = 180 - 140 = 40°
Congruent angles
congruent sides, so...
If the angles are all different, the sides are all different and the triangle is scalene. It is also acute because all angles are <90°.
Acute scalene triangle
Question 4
An exterior angle of a triangle = the sum of its remote interior angles.
what does this mean? In this case, it means that
4x + 10 = 50 + x + 5
The reason this works: <ACD is supplementary to <ACB, so they must both add up to 180. Likewise, the angles of the triangle <ACB, <BAC, and <ABC all add up to 180. Since both = 180, we can set them equal (substitution prop)
m<ACD + m<ACB = m<ACB + m<BAC + m<ABC
subtract m<ACB from each side
m<ACD = m<BAC + m<ABC
now let's solve our equation.
4x + 10 = 50 + x + 5
4x + 10 = 55 + x
subtract x from each side
3x + 10 = 55
subtract 10 from each side
3x = 45
divide by 3
x = 15
the measure of the interior angle = (x+5)° = (15+5)° = 20°
the measure of the exterior angle = (4x+10)° = (4(15)+10)° = (60+10)° = 70°
Question 5
As we can see from the diagram...
AB = CD, AD = BC
<ABD = <BDC, <ADB = <CBD
Also obvious from the reflexive prop...
BD = BD
And because the three angles add up to 180° and we know that two of them are the same, the third must be the same as well.
<A = <C
(three points don't need specified because only one angle is possible with this vertex)
Question 6
Because the angles in a triangle add up to 180°...
m<A + m<ADB + m<ABD = 180°
We know that m<A = 60°...
60 + m<ADB + m<ABD = 180°
We know that m<DBC = 70°, and from the diagram, <ADB is cong. to <DBC.
Therefore m<ADB = 70°.
60 + 70 m<ABD = 180°
130 + m<ABD = 180°
m<ABD = 50°
A triangle is:
either scalene, isoceles, or equilateral, and
either acute, right, or obtuse.
Question 1: The triangle is right b/c it has a right angle.
The triangle is isoceles because it has two congruent sides.
Right isoceles triangle
Question 2: The triangle is obtuse b/c it has an obtuse angle. (>90°)
The triangle is isoceles b/c it has two congruent sides.
Obtuse isoceles triangle
Question 3: All angles in a triangle add up to 180°. Let's find the 3rd angle.
180 - (80 + 60) = 180 - 140 = 40°
Congruent angles
If the angles are all different, the sides are all different and the triangle is scalene. It is also acute because all angles are <90°.
Acute scalene triangle
Question 4
An exterior angle of a triangle = the sum of its remote interior angles.
what does this mean? In this case, it means that
4x + 10 = 50 + x + 5
The reason this works: <ACD is supplementary to <ACB, so they must both add up to 180. Likewise, the angles of the triangle <ACB, <BAC, and <ABC all add up to 180. Since both = 180, we can set them equal (substitution prop)
m<ACD + m<ACB = m<ACB + m<BAC + m<ABC
subtract m<ACB from each side
m<ACD = m<BAC + m<ABC
now let's solve our equation.
4x + 10 = 50 + x + 5
4x + 10 = 55 + x
subtract x from each side
3x + 10 = 55
subtract 10 from each side
3x = 45
divide by 3
x = 15
the measure of the interior angle = (x+5)° = (15+5)° = 20°
the measure of the exterior angle = (4x+10)° = (4(15)+10)° = (60+10)° = 70°
Question 5
As we can see from the diagram...
AB = CD, AD = BC
<ABD = <BDC, <ADB = <CBD
Also obvious from the reflexive prop...
BD = BD
And because the three angles add up to 180° and we know that two of them are the same, the third must be the same as well.
<A = <C
(three points don't need specified because only one angle is possible with this vertex)
Question 6
Because the angles in a triangle add up to 180°...
m<A + m<ADB + m<ABD = 180°
We know that m<A = 60°...
60 + m<ADB + m<ABD = 180°
We know that m<DBC = 70°, and from the diagram, <ADB is cong. to <DBC.
Therefore m<ADB = 70°.
60 + 70 m<ABD = 180°
130 + m<ABD = 180°
m<ABD = 50°