Question

9 tell whether the following sides could form a triangle: 10 cm, 5 cm, 7 cm. show work to support your answer.
10 solve for x and uw. show all work.
(image of triangle uvw with uw = 5x, uv = 40, vw = 6x + 4, and uv ≅ vw)
x = ______
uw = ______
11 complete the flow proof
given: △def with right ∠f.
prove: ∠d and ∠e are complementary
(image of right triangle def with right angle at f and flow proof diagram, along with word bank: definition of complementary angles, substitution property, m∠f = 90°, △def with right ∠f, triangle sum theorem, m∠d + m∠e = 90°)
12 if the measures of two sides of a triangle are 5 yards and 9 yards, circle all the possible measures for the third side if the measure is an integer.
a. 5 yards b. 6 yards c. 10 yards d. 14 yards e. 4 yards f. 15 yards

Explanation:

Question 9

Step1: Recall Triangle Inequality Theorem

The Triangle Inequality Theorem states that for three side lengths \(a\), \(b\), and \(c\) to form a triangle, the sum of any two sides must be greater than the third side. So we need to check three inequalities: \(a + b>c\), \(a + c>b\), and \(b + c>a\).

Step2: Identify the sides

The given side lengths are \(10\) cm, \(5\) cm, and \(7\) cm. Let \(a = 5\), \(b = 7\), \(c = 10\).

Step3: Check the first inequality (\(a + b>c\))

Calculate \(5+7 = 12\). Since \(12>10\), this inequality holds.

Step4: Check the second inequality (\(a + c>b\))

Calculate \(5 + 10=15\). Since \(15>7\), this inequality holds.

Step5: Check the third inequality (\(b + c>a\))

Calculate \(7+10 = 17\). Since \(17>5\), this inequality holds.

Step1: Identify the isosceles triangle

In triangle \(UVW\), sides \(UV\) and \(VW\) are marked as equal (the tick marks), so \(UV=VW\). We know \(UV = 40\) and \(VW=6x + 4\), and \(UW = 5x\).

Step2: Set up the equation

Since \(UV = VW\), we set \(40=6x + 4\).

Step3: Solve for \(x\)

Subtract \(4\) from both sides: \(40-4=6x\), so \(36 = 6x\). Then divide both sides by \(6\): \(x=\frac{36}{6}=6\).

Step4: Find \(UW\)

Now that \(x = 6\), substitute \(x\) into the expression for \(UW\), which is \(5x\). So \(UW=5\times6 = 30\).

Step1: Fill in the "Given" box

The given is \(\triangle DEF\) with right \(\angle F\), so we put "\(\triangle DEF\) with right \(\angle F\)" in the first oval (Given).

Step2: Fill in the "Definition of a right \(\angle\)" box

By the definition of a right angle, \(m\angle F=90^{\circ}\), so we put \(m\angle F = 90^{\circ}\) in the second oval.

Step3: Fill in the reason for \(m\angle D + m\angle E+m\angle F = 180^{\circ}\)

This is by the Triangle Sum Theorem, which states that the sum of the interior angles of a triangle is \(180^{\circ}\), so the reason is "Triangle Sum Theorem".

Step4: Fill in the reason for \(m\angle D + m\angle E+90^{\circ}=180^{\circ}\)

We substitute \(m\angle F = 90^{\circ}\) into \(m\angle D + m\angle E+m\angle F = 180^{\circ}\), so the reason is "Substitution Property".

Step5: Fill in the box for Subtraction Property of Equality

Using the Subtraction Property of Equality, we subtract \(90^{\circ}\) from both sides of \(m\angle D + m\angle E+90^{\circ}=180^{\circ}\) to get \(m\angle D + m\angle E=90^{\circ}\).

Step6: Fill in the last box

By the Definition of Complementary Angles (two angles are complementary if their measures sum to \(90^{\circ}\)), \(\angle D\) and \(\angle E\) are complementary, so we put "Definition of Complementary Angles" as the reason.

Answer:

Yes, the sides \(10\) cm, \(5\) cm, and \(7\) cm can form a triangle because the sum of any two sides is greater than the third side ( \(5 + 7>10\), \(5+10>7\), and \(7 + 10>5\) ).

Question 10