Exterior angles of a triangle - Examples, Exercises and Solutions

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

A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In triangle $\triangle ABC$, we need to identify such a median from the diagram provided.

Step 1: Observe the diagram to identify the midpoint of each side.

Step 2: It is given that point $E$ is located on side $AC$. If $E$ is the midpoint of $AC$, then any line from a vertex to point $E$ would be a median.

Step 3: Check line segment $BE$. This line runs from vertex $B$ to point $E$.

Step 4: Since $E$ is labeled as the midpoint of $AC$, line $BE$ is the median of $\triangle ABC$ drawn to side $AC$.

Therefore, the median of the triangle is $BE$ for $AC$.

To solve this problem, we must identify which line segment in triangle ABC is the median.

First, review the definition: a median in a triangle connects a vertex to the midpoint of the opposite side. Now, in triangle ABC:

• Point A represents the vertex.
• Point E lies on line segment AB.
• Line segment EC needs to be checked to see if it connects vertex E to point C.

From the diagram, it appears that E is indeed the midpoint of side AB. Thus, line segment EC connects vertex C to this midpoint.

This fits the definition of a median, verifying that EC is the median line segment in triangle ABC.

Therefore, the solution to the problem is: $\text{EC}$.

A median in a triangle is a line segment connecting a vertex to the midpoint of the opposite side. Here, we need to find such a segment in triangle $\triangle ABC$.

Let's analyze the given conditions:

• $AD = \frac{1}{2}AB$: Point $D$ is the midpoint of $AB$.
• $BE = \frac{1}{2}EC$: Point $E$ is the midpoint of $EC$.

Given that $D$ is the midpoint of $AB$, if we consider the line segment $DC$, it starts from vertex $D$ and ends at $C$, passing through the midpoint of $AB$ (which is $D$), fulfilling the condition for a median.

Therefore, the line segment $DC$ is the median from vertex $A$ to side $BC$.

In summary, the correct answer is the segment $DC$.

To solve the problem of identifying the median of triangle $\triangle ABC$, we follow these steps:

• Step 1: Understand the Definition - A median of a triangle is a line segment that extends from a vertex to the midpoint of the opposite side.
• Step 2: Identify Potential Medians - Examine segments from each vertex to the opposite side. The diagram labels these connections.
• Step 3: Confirm the Median - Specifically check the segment EC in the context of the line segment from vertex $E$ to the side $AC$, and verify it reaches the midpoint of side $AC$.
• Step 4: Verify Against Options - Given choices allow us to consider which point-to-point connection adheres to our criterion for a median. EC is given as one of the choices.

Observation shows: From point $E$ (assumed from the label and position) that line extends directly to point $C$—a crucial diagonal opposite from considered midpoint indications, suggesting it cuts $AC$ evenly, classifying it as a median.

Upon reviewing the given choices, we see that segment $EC$ is listed. Confirming that $EC$ indeed meets at $C$, the midpoint of $AC$, validates that it is a true median.

Therefore, the correct median of $\triangle ABC$ is the segment $EC$.

To determine the median of triangle $ABC$, we need to identify the line that extends from one vertex to the midpoint of the opposite side.

• Step 1: Review the given line segments in the figure.
• Step 2: Recall that a median connects a vertex to the midpoint of the opposite side.
• Step 3: Examine each line in the context of $\triangle ABC$.

Let's consider each given line:

• Line $AF$ does not appear to connect to the midpoint of any side of the triangle directly.
• Line $DE$ is an internal line and does not serve as a median of the main triangle $ABC$.
• Line $FE$ is similar to $DE$, serving non-median purposes interior to another structure.
• Line $AG$ starts at vertex $A$ and extends to point $G$, lying on side $BC$. If $G$ is the midpoint of $BC$, then $AG$ qualifies as the median.

Verification: Point $G$ is positioned directly between points $B$ and $C$ along line $BC$, confirming its role as the midpoint.

Thus, the line $AG$ is indeed the median of triangle $ABC$ since it fulfills connecting vertex $A$ and the midpoint of side $BC$.

Therefore, the solution to the problem is $AG$ as the median of triangle $ABC$.

To determine the median of triangle ABC, we must identify a segment connecting a vertex of the triangle to the midpoint of the opposite side.

Examining the diagram, point F appears to be located on side AC. Given the configuration, point F divides side AC into two equal segments, which makes F the midpoint of AC.

Therefore, segment CF connects vertex C to the midpoint F of side AC. This characteristic aligns with the definition of a median in a triangle.

Hence, the median of triangle ABC is $CF$.

In this problem, we must determine if any of the line segments drawn within triangle ABC represent a median. A median is defined as a line segment extending from a vertex to the midpoint of the opposite side.

Upon examining the geometry of triangle ABC presented in the diagram:

• The segment extending from A to the base BC and those depicted from B or C should be checked if they connect to a midpoint on the opposite side.
• To be considered a median, a line from a vertex must bisect the opposite side into two equal lengths.

None of the segments drawn directly bisect the opposite sides they connect to, as evidenced by either lack of midpoint marking or unequal line segment sections along BC, CA, or AB.

Therefore, after careful inspection, there is no median shown in the given diagram.

To solve this problem, we apply the definition of a median in a triangle. A median is a line segment drawn from a vertex to the midpoint of the opposite side. In the diagram of the triangle ABC:

• Line $AD$ originates from vertex $A$ and is directed towards point $D$ on side $BC$.
• We need to check if $D$ is the midpoint of $BC$.
• Given that $AD$ meets the definition of a median by dividing $BC$ into two equal segments, it is indeed the median.

After evaluating the possible choices:

• Choice 1: $AD$ is a line from $A$ to the midpoint of $BC$.
• Choice 2: $AE$ doesn't bisect any side.
• Choice 3: $EC$ is not a median.
• Choice 4: $AC$ does not connect a vertex to a midpoint of the opposite side.

Therefore, the solution to the problem is that line segment AD is the median of triangle ABC.

To identify the median in triangle $ABC$, we will utilize the definition of a median: it is the line segment extending from a vertex of the triangle to the midpoint of the opposite side.

In the diagram, triangle $ABC$ is formed with vertices $A$, $B$, and $C$. We need to identify which of the segments is drawn from a vertex and intersects the opposite side at its midpoint.

Examine segment $FC$:

• $F$ appears to be a midpoint of side $AB$ of the triangle $ABC$.
• Line segment $FC$ originates from vertex $C$ and extends to $F$.

The segment $FC$ meets the criteria for a median as it connects vertex $C$ to the midpoint of $AB$.

Therefore, we conclude that the median of triangle $ABC$ is FC.

To solve this problem, let's clarify the role of AB in the context of triangle ABC by analyzing its diagram:

• Step 1: Identify the vertices of the triangle. According to the diagram, the vertices of the triangle are points labeled A, B, and C.
• Step 2: Determine the sides of the triangle. In any triangle, the sides are the segments connecting pairs of distinct vertices.
• Step 3: Identify AB as a line segment connecting vertex A and vertex B, labeled directly in the diagram.

Considering these steps, line segment AB connects vertex A with vertex B, and hence, forms one of the sides of the triangle ABC. Therefore, AB is indeed a side of triangle ABC as shown in the diagram.

The conclusion here is solidly supported by our observation of the given triangle. Thus, the statement that AB is a side of the triangle ABC is True.