Triangle Angles Practice Problems - Sum Theorem Exercises

To solve the problem of determining whether DE is not a side in any of the triangles, we will methodically identify the triangles present in the diagram and examine their sides:

• Identify triangles in the diagram. The diagram presented forms a right-angled triangle ABC with additional lines forming smaller triangles within.
• Triangles formed: Triangle ABC (major triangle), Triangle ABD, Triangle BEC, and Triangle DBE.
• Let's examine the sides of these triangles:
• Triangle ABC has sides AB, BC, and CA.
• Triangle ABD has sides AB, BD, and DA.
• Triangle BEC has sides BE, EC, and CB.
• Triangle DBE has sides DB, BE, and ED.
• Notice that while point D is used, the segment DE is only part of line BE and isn't listed as a direct side of any triangle.

Therefore, the claim that DE is not a side in any of the triangles is indeed correct.

Hence, the answer is True.

Since line segment DE does not correspond to a full side of any of the triangles present within the given geometry, we conclude that the statement “DE is a side in one of the triangles” is Not true.

To find the size of the missing angle, we will use the property that the sum of angles on a straight line is $180^\circ$. Given that one angle is $80^\circ$, we can calculate the missing angle using the following steps:

• Step 1: Recognize that the given angle $\alpha = 80^\circ$ and the missing angle $\beta$ form a straight line.
• Step 2: Use the angle sum property for a straight line: $\alpha + \beta = 180^\circ$
• Step 3: Substitute the known value: $80^\circ + \beta = 180^\circ$
• Step 4: Solve for the missing angle $\beta$: $\beta = 180^\circ - 80^\circ = 100^\circ$

Therefore, the size of the missing angle is $100^\circ$.

To solve the problem of identifying to which side of triangle $ABC$ the median and the altitude are drawn, let's analyze the diagram given for triangle $ABC$.

• We acknowledge that a median is a line segment drawn from a vertex to the midpoint of the opposite side. An altitude is a line segment drawn from a vertex perpendicular to the opposite side.
• Upon reviewing the diagram of triangle $ABC$, line segment $AD$ is a reference term. It appears to meet point $C$ in the middle, suggesting it's a median, but it also forms right angles suggesting it is an altitude.
• Given the placement and orientation of $AD$, it is perpendicular to line $BC$ (the opposite base for the median from $A$). Therefore, this line is both the median and the altitude to side $BC$.

Thus, the side to which both the median and the altitude are drawn is BC.

Therefore, the correct answer to the problem is the side $BC$, corresponding with choice $\text{Option 2: BC}$.

To solve this problem, we need to identify the median in triangle ABC:

• Step 1: Recall the definition of a median. A median is a line segment drawn from a vertex to the midpoint of the opposite side.
• Step 2: Begin by evaluating each line segment based on the definition.
• Step 3: Identify points on triangle ABC:
• AD is from A to a point on BC.
• BE is from B to a point on AC.
• FC is from F to a point on AB.
• Step 4: Determine if these points (D, E, F) are midpoints:
• Since BE connects B to E, and E is indicated to be the midpoint of segment AC (as shown), BE is the median.
• AD and FC, by visual inspection, do not connect to midpoints on BC or AB respectively.

Therefore, the line segment that represents the median is $BE$.

Thus, the correct answer is: BE